Question 15963: Quotient Multiplication Must Respect the Same Ideal

Original question and givens

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]).

• G001 — 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]).

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(16, 4)

quotient ring parent modulus divided by ideal generator = 4

same declared parent Z_16;generator d=4;n/d=4;ideal multiples=[0, 4, 8, 12]

I=kernel of residue map Z_16 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, 8, 9, 10, 11, 12, 13, 14, 15];same ideal I=[0, 4, 8, 12]

all additive cosets in representative order0..d-1=[[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]];quotient size=4;no field conclusion

first input coset exact quotient fraction := div(12, 4)

first input coset exact quotient fraction = 3

first input coset:integer value=12;positive modulus=4;exact ratio=3

floor quotient=3;quotient<=ratio<quotient+1

first input coset quotient multiple := mul(3, 4)

first input coset quotient multiple = 12

first input coset canonical nonnegative residue := sub(12, 12)

first input coset canonical nonnegative residue = 0

second input coset exact quotient fraction := div(9, 4)

second input coset exact quotient fraction = 9/4

second input coset:integer value=9;positive modulus=4;exact ratio=9/4

floor quotient=2;quotient<=ratio<quotient+1

second input coset quotient multiple := mul(2, 4)

second input coset quotient multiple = 8

second input coset canonical nonnegative residue := sub(9, 8)

second input coset canonical nonnegative residue = 1

selected quotient operation integer multiplication := mul(12, 9)

selected quotient operation integer multiplication = 108

selected quotient operation parent mod n exact quotient fraction := div(108, 16)

selected quotient operation parent mod n exact quotient fraction = 27/4

selected quotient operation parent mod n:integer value=108;positive modulus=16;exact ratio=27/4

floor quotient=6;quotient<=ratio<quotient+1

selected quotient operation parent mod n quotient multiple := mul(6, 16)

selected quotient operation parent mod n quotient multiple = 96

selected quotient operation parent mod n canonical nonnegative residue := sub(108, 96)

selected quotient operation parent mod n canonical nonnegative residue = 12

selected quotient operation quotient class mod d exact quotient fraction := div(12, 4)

selected quotient operation quotient class mod d exact quotient fraction = 3

selected quotient operation quotient class mod d:integer value=12;positive modulus=4;exact ratio=3

floor quotient=3;quotient<=ratio<quotient+1

selected quotient operation quotient class mod d quotient multiple := mul(3, 4)

selected quotient operation quotient class mod d quotient multiple = 12

selected quotient operation quotient class mod d canonical nonnegative residue := sub(12, 12)

selected quotient operation quotient class mod d canonical nonnegative residue = 0

selected quotient operation:original parent representatives=[12, 9];integer result=108;parent modulus=16

parent result=12;quotient class label=0;class modulus=4

same parent Z_16;same proper two-sided ideal I=[0, 4, 8, 12];input class labels=[0, 1]

quotient operation law=[a][b]=[ab];result class label=0

Reverse checks

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute selected quotient operation quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute selected quotient operation quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute selected quotient operation quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute selected quotient operation parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute selected quotient operation parent mod n quotient multiple with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute selected quotient operation parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute selected quotient operation integer multiplication with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute second input coset canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute second input coset quotient multiple with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute second input coset exact quotient fraction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute first input coset canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute first input coset quotient multiple with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute first input coset exact quotient fraction with exact rational arithmetic, not rounded decimals.
• 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, 1, 12, 0, [a][b]=[ab])

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.