第 15953 題：有限商環

原題與已知條件

Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={"n":8,"d":4,"a":2,"b":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。

• G001 — Z_n與proper雙邊理想的全部加法cosets。操作=quotring-multiply；資料={"n":8,"d":4,"a":2,"b":5}。請依 [class_a,class_b,parent_result,result_class,law] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-finite-quotient-ring

逐步推導

quotient ring parent modulus divided by ideal generator := div(8, 4)

quotient ring parent modulus divided by ideal generator = 2

same declared parent Z_8;generator d=4;n/d=2;ideal multiples=[0, 4]

I=kernel of residue map Z_8 to Z_4;proper two-sided ideal=是;zero ideal allowed when d=n

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

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

first input coset exact quotient fraction := div(2, 4)

first input coset exact quotient fraction = 1/2

first input coset:integer value=2;positive modulus=4;exact ratio=1/2

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

first input coset quotient multiple := mul(0, 4)

first input coset quotient multiple = 0

first input coset canonical nonnegative residue := sub(2, 0)

first input coset canonical nonnegative residue = 2

second input coset exact quotient fraction := div(5, 4)

second input coset exact quotient fraction = 5/4

second input coset:integer value=5;positive modulus=4;exact ratio=5/4

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

second input coset quotient multiple := mul(1, 4)

second input coset quotient multiple = 4

second input coset canonical nonnegative residue := sub(5, 4)

second input coset canonical nonnegative residue = 1

selected quotient operation integer multiplication := mul(2, 5)

selected quotient operation integer multiplication = 10

selected quotient operation parent mod n exact quotient fraction := div(10, 8)

selected quotient operation parent mod n exact quotient fraction = 5/4

selected quotient operation parent mod n:integer value=10;positive modulus=8;exact ratio=5/4

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

selected quotient operation parent mod n quotient multiple := mul(1, 8)

selected quotient operation parent mod n quotient multiple = 8

selected quotient operation parent mod n canonical nonnegative residue := sub(10, 8)

selected quotient operation parent mod n canonical nonnegative residue = 2

selected quotient operation quotient class mod d exact quotient fraction := div(2, 4)

selected quotient operation quotient class mod d exact quotient fraction = 1/2

selected quotient operation quotient class mod d:integer value=2;positive modulus=4;exact ratio=1/2

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

selected quotient operation quotient class mod d quotient multiple := mul(0, 4)

selected quotient operation quotient class mod d quotient multiple = 0

selected quotient operation quotient class mod d canonical nonnegative residue := sub(2, 0)

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

selected quotient operation:original parent representatives=[2, 5];integer result=10;parent modulus=8

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

same parent Z_8;same proper two-sided ideal I=[0, 4];input class labels=[2, 1]

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

逐步反向核對

• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 selected quotient operation quotient class mod d canonical nonnegative residue，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 selected quotient operation quotient class mod d quotient multiple，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 selected quotient operation quotient class mod d exact quotient fraction，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 selected quotient operation parent mod n canonical nonnegative residue，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 selected quotient operation parent mod n quotient multiple，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 selected quotient operation parent mod n exact quotient fraction，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 selected quotient operation integer multiplication，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 second input coset canonical nonnegative residue，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 second input coset quotient multiple，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 second input coset exact quotient fraction，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 first input coset canonical nonnegative residue，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 first input coset quotient multiple，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 first input coset exact quotient fraction，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 以精確有理數重算 quotient ring parent modulus divided by ideal generator，不以小數近似替代。

結論

(2, 1, 2, 2, [a][b]=[ab])

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。