# Question 15947: Build the Quotient from Full Additive Cosets [Return to catalogue page 80](https://ocme.evemisslab.com/en/questions/page/80/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations/#q-20261014-advance-20261031-r04-finite-quotient-ring-0022) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0030-15947 · q-20261014-advance-20261031-r04-finite-quotient-ring-0022 ## Original question and givens 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). - G001 — 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). ## 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](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-ring/) ## Step-by-step derivation S001 · calculationBefore quotient ring parent modulus divided by ideal generator := div(18, 3) After quotient ring parent modulus divided by ideal generator = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore same declared parent Z_18;generator d=3;n/d=6;ideal multiples=[0, 3, 6, 9, 12, 15] After I=kernel of residue map Z_18 to Z_3;proper two-sided ideal=Yes;zero ideal allowed when d=n Conditions: none Since d divides n,changing an integer representative by n does not change mod d.The map π:Z_n→Z_d is well-defined and preserves addition and multiplication.Its kernel consists of multiples of d,closed under differences and multiplication on either side by parent elements.With d≥2 it excludes1 and is proper;d=n gives the allowed zero ideal. Step identifiers and dependenciesRule: quotient_ring_actual_parent_proper_two_sided_ideal · Uses: G001, S001 S003 · impliesBefore complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];same ideal I=[0, 3, 6, 9, 12, 15] After all additive cosets in representative order0..d-1=[[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]];quotient size=3;no field conclusion Conditions: none List every parent element congruent mod d in each additive coset,ordered by representatives0 through d−1.They are disjoint and cover Z_n.Quotient elements are whole cosets;class labels are indices only,and a finite quotient ring is not automatically a field. Step identifiers and dependenciesRule: quotient_ring_full_cosets_not_representative_carrier · Uses: G001, S002 ## 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, 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) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: db0d07c1cf03e67a37e1a20013df57e52c86acb2f1fb6ba0e421d75dce618c6e - candidate record SHA-256: 7cffe9118ed7057997334546018dc5fc7295015da0bf33031c2f9ce21b3e3f16 - semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc - question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0022.json)