Question 15905: Add in the Parent Ring, Then Take the Quotient Class

Original question and givens

Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, 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=4, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

• G001 — Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, 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=4, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

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(12, 6)

quotient ring parent modulus divided by ideal generator = 2

same declared parent Z_12;generator d=6;n/d=2;ideal multiples=[0, 6]

I=kernel of residue map Z_12 to Z_6;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];same ideal I=[0, 6]

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

first input coset exact quotient fraction := div(4, 6)

first input coset exact quotient fraction = 2/3

first input coset:integer value=4;positive modulus=6;exact ratio=2/3

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

first input coset quotient multiple := mul(0, 6)

first input coset quotient multiple = 0

first input coset canonical nonnegative residue := sub(4, 0)

first input coset canonical nonnegative residue = 4

second input coset exact quotient fraction := div(9, 6)

second input coset exact quotient fraction = 3/2

second input coset:integer value=9;positive modulus=6;exact ratio=3/2

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

second input coset quotient multiple := mul(1, 6)

second input coset quotient multiple = 6

second input coset canonical nonnegative residue := sub(9, 6)

second input coset canonical nonnegative residue = 3

selected quotient operation integer addition := sum(4, 9)

selected quotient operation integer addition = 13

selected quotient operation parent mod n exact quotient fraction := div(13, 12)

selected quotient operation parent mod n exact quotient fraction = 13/12

selected quotient operation parent mod n:integer value=13;positive modulus=12;exact ratio=13/12

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

selected quotient operation parent mod n quotient multiple := mul(1, 12)

selected quotient operation parent mod n quotient multiple = 12

selected quotient operation parent mod n canonical nonnegative residue := sub(13, 12)

selected quotient operation parent mod n canonical nonnegative residue = 1

selected quotient operation quotient class mod d exact quotient fraction := div(1, 6)

selected quotient operation quotient class mod d exact quotient fraction = 1/6

selected quotient operation quotient class mod d:integer value=1;positive modulus=6;exact ratio=1/6

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

selected quotient operation quotient class mod d quotient multiple := mul(0, 6)

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=[4, 9];integer result=13;parent modulus=12

parent result=1;quotient class label=1;class modulus=6

same parent Z_12;same proper two-sided ideal I=[0, 6];input class labels=[4, 3]

quotient operation law=[a]+[b]=[a+b];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 addition 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

(4, 3, 1, 1, [a]+[b]=[a+b])

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.