# Proper-Ideal Cosets and Representative-Independent Quotient Operations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Build the Quotient from Full Additive Cosets

Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0001/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 2. Build the Quotient from Full Additive Cosets

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0002/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 3. Build the Quotient from Full Additive Cosets

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0003/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 4. Build the Quotient from Full Additive Cosets

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0004/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 5. Build the Quotient from Full Additive Cosets

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0005/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 6. Build the Quotient from Full Additive Cosets

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0006/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 7. Build the Quotient from Full Additive Cosets

Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0007/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 8. Build the Quotient from Full Additive Cosets

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0008/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 9. Build the Quotient from Full Additive Cosets

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0009/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 10. Build the Quotient from Full Additive Cosets

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0010/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 11. Build the Quotient from Full Additive Cosets

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0011/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 12. Build the Quotient from Full Additive Cosets

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0012/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 13. Build the Quotient from Full Additive Cosets

Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0013/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 14. Build the Quotient from Full Additive Cosets

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0014/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 15. Build the Quotient from Full Additive Cosets

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0015/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 16. Build the Quotient from Full Additive Cosets

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0016/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 17. Build the Quotient from Full Additive Cosets

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0017/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 18. Build the Quotient from Full Additive Cosets

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0018/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 19. Build the Quotient from Full Additive Cosets

Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0019/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 20. Build the Quotient from Full Additive Cosets

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0020/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 21. Build the Quotient from Full Additive Cosets

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0021/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 22. Build the Quotient from Full Additive Cosets

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0022/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 23. Build the Quotient from Full Additive Cosets

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0023/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 24. Build the Quotient from Full Additive Cosets

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0024/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 25. Build the Quotient from Full Additive Cosets

Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).

View answer and explanation([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)

List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0025/)

Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate





## 26. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 1, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0026/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 27. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 0, 4, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0027/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 28. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(2, 1, 7, 3, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 7, 3, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0028/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 29. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 2, 0, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0029/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 30. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(4, 3, 1, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (4, 3, 1, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0030/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 31. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 1, 4, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 4, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0031/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 32. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 1, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0032/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 33. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 3, 10, 2, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 3, 10, 2, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0033/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 34. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 2, 10, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0034/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 35. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 1, 10, 4, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 1, 10, 4, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0035/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 36. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 7, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 7, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0036/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 37. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(2, 2, 10, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 2, 10, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0037/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 38. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 5, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 5, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0038/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 39. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 2, 0, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0039/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 40. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(2, 5, 19, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 5, 19, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0040/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 41. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 1, 6, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 6, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0041/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 42. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 0, 4, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0042/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 43. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 3, 12, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 12, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0043/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 44. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 2, 5, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 5, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0044/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 45. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(1, 3, 28, 4, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 28, 4, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0045/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 46. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 1, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0046/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 47. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 10, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 10, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0047/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 48. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(2, 1, 19, 3, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 19, 3, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0048/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 49. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(3, 2, 10, 0, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0049/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 50. Add in the Parent Ring, Then Take the Quotient Class

Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

View answer and explanation(0, 1, 1, 1, [a]+[b]=[a+b])

Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0050/)

Quotient ring of a finite ring · qf-mod31-quotring-add · candidate





## 51. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 0, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 0, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0051/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 52. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 0, 3, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0052/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 53. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(2, 1, 2, 2, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 2, 2, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0053/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 54. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 2, 1, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0054/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 55. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(4, 3, 0, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (4, 3, 0, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0055/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 56. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 1, 1, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 1, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0056/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 57. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 6, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 6, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0057/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 58. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 3, 9, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 3, 9, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0058/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 59. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 2, 1, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0059/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 60. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 1, 9, 3, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 1, 9, 3, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0060/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 61. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 2, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 2, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0061/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 62. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(2, 2, 1, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 2, 1, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0062/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 63. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 12, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 12, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0063/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 64. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 2, 11, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 11, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0064/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 65. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(2, 5, 22, 4, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 5, 22, 4, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0065/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 66. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 1, 5, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 5, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0066/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 67. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 0, 3, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0067/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 68. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 3, 15, 3, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 15, 3, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0068/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 69. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 2, 16, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 16, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0069/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 70. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(1, 3, 21, 3, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 21, 3, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0070/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 71. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 4, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 4, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0071/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 72. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 3, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 3, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0072/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 73. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(2, 1, 6, 2, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 6, 2, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0073/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 74. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(3, 2, 1, 1, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0074/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 75. Quotient Multiplication Must Respect the Same Ideal

Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

View answer and explanation(0, 1, 24, 0, [a][b]=[ab])

Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 24, 0, [a][b]=[ab]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0075/)

Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate





## 76. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0076/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 77. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0077/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 78. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0078/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 79. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0079/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 80. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=4, b=3, a_prime=10, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0080/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 81. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0081/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 82. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0082/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 83. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=3, a_prime=7, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0083/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 84. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0084/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 85. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=1, a_prime=9, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0085/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 86. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0086/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 87. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=2, a_prime=5, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0087/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 88. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=4, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0088/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 89. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0089/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 90. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=5, a_prime=8, b_prime=11 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0090/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 91. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0091/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 92. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0092/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 93. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=5, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0093/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 94. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0094/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 95. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=7, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0095/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 96. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0096/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 97. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0097/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 98. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0098/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 99. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0099/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate





## 100. Changing Both Representatives Must Preserve Both Operations

Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=6, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).

View answer and explanation([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes)

Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0100/)

Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations/validation-report.json)