Question 15973: Quotient Multiplication Must Respect the Same Ideal

Original question and givens

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]).

• G001 — 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]).

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(24, 4)

quotient ring parent modulus divided by ideal generator = 6

same declared parent Z_24;generator d=4;n/d=6;ideal multiples=[0, 4, 8, 12, 16, 20]

I=kernel of residue map Z_24 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, 16, 17, 18, 19, 20, 21, 22, 23];same ideal I=[0, 4, 8, 12, 16, 20]

all additive cosets in representative order0..d-1=[[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]];quotient size=4;no field conclusion

first input coset exact quotient fraction := div(22, 4)

first input coset exact quotient fraction = 11/2

first input coset:integer value=22;positive modulus=4;exact ratio=11/2

floor quotient=5;quotient<=ratio<quotient+1

first input coset quotient multiple := mul(5, 4)

first input coset quotient multiple = 20

first input coset canonical nonnegative residue := sub(22, 20)

first input coset canonical nonnegative residue = 2

second input coset exact quotient fraction := div(21, 4)

second input coset exact quotient fraction = 21/4

second input coset:integer value=21;positive modulus=4;exact ratio=21/4

floor quotient=5;quotient<=ratio<quotient+1

second input coset quotient multiple := mul(5, 4)

second input coset quotient multiple = 20

second input coset canonical nonnegative residue := sub(21, 20)

second input coset canonical nonnegative residue = 1

selected quotient operation integer multiplication := mul(22, 21)

selected quotient operation integer multiplication = 462

selected quotient operation parent mod n exact quotient fraction := div(462, 24)

selected quotient operation parent mod n exact quotient fraction = 77/4

selected quotient operation parent mod n:integer value=462;positive modulus=24;exact ratio=77/4

floor quotient=19;quotient<=ratio<quotient+1

selected quotient operation parent mod n quotient multiple := mul(19, 24)

selected quotient operation parent mod n quotient multiple = 456

selected quotient operation parent mod n canonical nonnegative residue := sub(462, 456)

selected quotient operation parent mod n canonical nonnegative residue = 6

selected quotient operation quotient class mod d exact quotient fraction := div(6, 4)

selected quotient operation quotient class mod d exact quotient fraction = 3/2

selected quotient operation quotient class mod d:integer value=6;positive modulus=4;exact ratio=3/2

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

selected quotient operation quotient class mod d quotient multiple := mul(1, 4)

selected quotient operation quotient class mod d quotient multiple = 4

selected quotient operation quotient class mod d canonical nonnegative residue := sub(6, 4)

selected quotient operation quotient class mod d canonical nonnegative residue = 2

selected quotient operation:original parent representatives=[22, 21];integer result=462;parent modulus=24

parent result=6;quotient class label=2;class modulus=4

same parent Z_24;same proper two-sided ideal I=[0, 4, 8, 12, 16, 20];input class labels=[2, 1]

quotient operation law=[a][b]=[ab];result class label=2

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

(2, 1, 6, 2, [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.