# Question 15951: Quotient Multiplication Must Respect the Same Ideal [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-0051) 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. R0030-15951 · q-20261014-advance-20261031-r04-finite-quotient-ring-0051 ## Original question and givens 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]). - G001 — 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]). ## 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(4, 2) After quotient ring parent modulus divided by ideal generator = 2 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_4;generator d=2;n/d=2;ideal multiples=[0, 2] After I=kernel of residue map Z_4 to Z_2;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];same ideal I=[0, 2] After all additive cosets in representative order0..d-1=[[0, 2], [1, 3]];quotient size=2;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 S004 · calculationBefore first input coset exact quotient fraction := div(0, 2) After first input coset exact quotient fraction = 0 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, S003 S005 · impliesBefore first input coset:integer value=0;positive modulus=2;exact ratio=0 After floor quotient=0;quotient<=ratio