Question 15918: Add in the Parent Ring, Then Take the Quotient Class

Original question and givens

Use the parent ring Z_20 with residues 0 through 19 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=17, b=15. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).

• G001 — Use the parent ring Z_20 with residues 0 through 19 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=17, b=15. 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(20, 4)

quotient ring parent modulus divided by ideal generator = 5

same declared parent Z_20;generator d=4;n/d=5;ideal multiples=[0, 4, 8, 12, 16]

I=kernel of residue map Z_20 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];same ideal I=[0, 4, 8, 12, 16]

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

first input coset exact quotient fraction := div(17, 4)

first input coset exact quotient fraction = 17/4

first input coset:integer value=17;positive modulus=4;exact ratio=17/4

floor quotient=4;quotient<=ratio<quotient+1

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

first input coset quotient multiple = 16

first input coset canonical nonnegative residue := sub(17, 16)

first input coset canonical nonnegative residue = 1

second input coset exact quotient fraction := div(15, 4)

second input coset exact quotient fraction = 15/4

second input coset:integer value=15;positive modulus=4;exact ratio=15/4

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

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

second input coset quotient multiple = 12

second input coset canonical nonnegative residue := sub(15, 12)

second input coset canonical nonnegative residue = 3

selected quotient operation integer addition := sum(17, 15)

selected quotient operation integer addition = 32

selected quotient operation parent mod n exact quotient fraction := div(32, 20)

selected quotient operation parent mod n exact quotient fraction = 8/5

selected quotient operation parent mod n:integer value=32;positive modulus=20;exact ratio=8/5

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

selected quotient operation parent mod n quotient multiple := mul(1, 20)

selected quotient operation parent mod n quotient multiple = 20

selected quotient operation parent mod n canonical nonnegative residue := sub(32, 20)

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=[17, 15];integer result=32;parent modulus=20

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

same parent Z_20;same proper two-sided ideal I=[0, 4, 8, 12, 16];input class labels=[1, 3]

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

(1, 3, 12, 0, [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.