Question 15928: Build the Quotient from Full Additive Cosets

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-quotient-ring

Step-by-step derivation

quotient ring parent modulus divided by ideal generator := div(8, 4)

quotient ring parent modulus divided by ideal generator = 2

same declared parent Z_8;generator d=4;n/d=2;ideal multiples=[0, 4]

I=kernel of residue map Z_8 to Z_4;proper two-sided ideal=Yes;zero ideal allowed when d=n

complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7];same ideal I=[0, 4]

all additive cosets in representative order0..d-1=[[0, 4], [1, 5], [2, 6], [3, 7]];quotient size=4;no field conclusion

Reverse checks

• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute quotient ring parent modulus divided by ideal generator with exact rational arithmetic, not rounded decimals.

Result

([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.