Question 15954: Quotient Multiplication Must Respect the Same Ideal

Original question and givens

Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, 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=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).

• G001 — Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, 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=3, b=7. 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(10, 5)

quotient ring parent modulus divided by ideal generator = 2

same declared parent Z_10;generator d=5;n/d=2;ideal multiples=[0, 5]

I=kernel of residue map Z_10 to Z_5;proper two-sided ideal=Yes;zero ideal allowed when d=n

complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];same ideal I=[0, 5]

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

first input coset exact quotient fraction := div(3, 5)

first input coset exact quotient fraction = 3/5

first input coset:integer value=3;positive modulus=5;exact ratio=3/5

floor quotient=0;quotient<=ratio<quotient+1

first input coset quotient multiple := mul(0, 5)

first input coset quotient multiple = 0

first input coset canonical nonnegative residue := sub(3, 0)

first input coset canonical nonnegative residue = 3

second input coset exact quotient fraction := div(7, 5)

second input coset exact quotient fraction = 7/5

second input coset:integer value=7;positive modulus=5;exact ratio=7/5

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

second input coset quotient multiple := mul(1, 5)

second input coset quotient multiple = 5

second input coset canonical nonnegative residue := sub(7, 5)

second input coset canonical nonnegative residue = 2

selected quotient operation integer multiplication := mul(3, 7)

selected quotient operation integer multiplication = 21

selected quotient operation parent mod n exact quotient fraction := div(21, 10)

selected quotient operation parent mod n exact quotient fraction = 21/10

selected quotient operation parent mod n:integer value=21;positive modulus=10;exact ratio=21/10

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

selected quotient operation parent mod n quotient multiple := mul(2, 10)

selected quotient operation parent mod n quotient multiple = 20

selected quotient operation parent mod n canonical nonnegative residue := sub(21, 20)

selected quotient operation parent mod n canonical nonnegative residue = 1

selected quotient operation quotient class mod d exact quotient fraction := div(1, 5)

selected quotient operation quotient class mod d exact quotient fraction = 1/5

selected quotient operation quotient class mod d:integer value=1;positive modulus=5;exact ratio=1/5

floor quotient=0;quotient<=ratio<quotient+1

selected quotient operation quotient class mod d quotient multiple := mul(0, 5)

selected quotient operation quotient class mod d quotient multiple = 0

selected quotient operation quotient class mod d canonical nonnegative residue := sub(1, 0)

selected quotient operation quotient class mod d canonical nonnegative residue = 1

selected quotient operation:original parent representatives=[3, 7];integer result=21;parent modulus=10

parent result=1;quotient class label=1;class modulus=5

same parent Z_10;same proper two-sided ideal I=[0, 5];input class labels=[3, 2]

quotient operation law=[a][b]=[ab];result class label=1

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

(3, 2, 1, 1, [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.