第 15972 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":18,"d":3,"a":3,"b":7}。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":18,"d":3,"a":3,"b":7}。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(18, 3)

quotient ring parent modulus divided by ideal generator = 6

same declared parent Z_18;generator d=3;n/d=6;ideal multiples=[0, 3, 6, 9, 12, 15]

I=kernel of residue map Z_18 to Z_3;proper two-sided ideal=是;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];same ideal I=[0, 3, 6, 9, 12, 15]

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

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

first input coset exact quotient fraction = 1

first input coset:integer value=3;positive modulus=3;exact ratio=1

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

first input coset quotient multiple := mul(1, 3)

first input coset quotient multiple = 3

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

first input coset canonical nonnegative residue = 0

second input coset exact quotient fraction := div(7, 3)

second input coset exact quotient fraction = 7/3

second input coset:integer value=7;positive modulus=3;exact ratio=7/3

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

second input coset quotient multiple := mul(2, 3)

second input coset quotient multiple = 6

second input coset canonical nonnegative residue := sub(7, 6)

second input coset canonical nonnegative residue = 1

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, 18)

selected quotient operation parent mod n exact quotient fraction = 7/6

selected quotient operation parent mod n:integer value=21;positive modulus=18;exact ratio=7/6

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

selected quotient operation parent mod n quotient multiple := mul(1, 18)

selected quotient operation parent mod n quotient multiple = 18

selected quotient operation parent mod n canonical nonnegative residue := sub(21, 18)

selected quotient operation parent mod n canonical nonnegative residue = 3

selected quotient operation quotient class mod d exact quotient fraction := div(3, 3)

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

selected quotient operation quotient class mod d:integer value=3;positive modulus=3;exact ratio=1

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

selected quotient operation quotient class mod d quotient multiple := mul(1, 3)

selected quotient operation quotient class mod d quotient multiple = 3

selected quotient operation quotient class mod d canonical nonnegative residue := sub(3, 3)

selected quotient operation quotient class mod d canonical nonnegative residue = 0

selected quotient operation:original parent representatives=[3, 7];integer result=21;parent modulus=18

parent result=3;quotient class label=0;class modulus=3

same parent Z_18;same proper two-sided ideal I=[0, 3, 6, 9, 12, 15];input class labels=[0, 1]

quotient operation law=[a][b]=[ab];result class label=0

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

(0, 1, 3, 0, [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。