第 15981 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-well-defined；資料={"n":6,"d":2,"a":1,"b":1,"a_prime":3,"b_prime":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 欄序回答。

• G001 — Z_n與proper雙邊理想的全部加法cosets。操作=quotring-well-defined；資料={"n":6,"d":2,"a":1,"b":1,"a_prime":3,"b_prime":3}。請依 [representatives,addition_classes,multiplication_classes,addition_well_defined,multiplication_well_defined,quotient_operations_well_defined] 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(6, 2)

quotient ring parent modulus divided by ideal generator = 3

same declared parent Z_6;generator d=2;n/d=3;ideal multiples=[0, 2, 4]

I=kernel of residue map Z_6 to Z_2;proper two-sided ideal=是;zero ideal allowed when d=n

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

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

representative slot0 input class exact quotient fraction := div(1, 2)

representative slot0 input class exact quotient fraction = 1/2

representative slot0 input class:integer value=1;positive modulus=2;exact ratio=1/2

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

representative slot0 input class quotient multiple := mul(0, 2)

representative slot0 input class quotient multiple = 0

representative slot0 input class canonical nonnegative residue := sub(1, 0)

representative slot0 input class canonical nonnegative residue = 1

representative slot1 input class exact quotient fraction := div(1, 2)

representative slot1 input class exact quotient fraction = 1/2

representative slot1 input class:integer value=1;positive modulus=2;exact ratio=1/2

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

representative slot1 input class quotient multiple := mul(0, 2)

representative slot1 input class quotient multiple = 0

representative slot1 input class canonical nonnegative residue := sub(1, 0)

representative slot1 input class canonical nonnegative residue = 1

representative slot2 input class exact quotient fraction := div(3, 2)

representative slot2 input class exact quotient fraction = 3/2

representative slot2 input class:integer value=3;positive modulus=2;exact ratio=3/2

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

representative slot2 input class quotient multiple := mul(1, 2)

representative slot2 input class quotient multiple = 2

representative slot2 input class canonical nonnegative residue := sub(3, 2)

representative slot2 input class canonical nonnegative residue = 1

representative slot3 input class exact quotient fraction := div(3, 2)

representative slot3 input class exact quotient fraction = 3/2

representative slot3 input class:integer value=3;positive modulus=2;exact ratio=3/2

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

representative slot3 input class quotient multiple := mul(1, 2)

representative slot3 input class quotient multiple = 2

representative slot3 input class canonical nonnegative residue := sub(3, 2)

representative slot3 input class canonical nonnegative residue = 1

first representative change := sub(1, 3)

first representative change = -2

second representative change := sub(1, 3)

second representative change = -2

first representative change divided by d := div(-2, 2)

first representative change divided by d = -1

second representative change divided by d := div(-2, 2)

second representative change divided by d = -1

original representative choices=[1, 1, 3, 3];class labels in slot order=[1, 1, 1, 1];changes a-a_prime=-2;b-b_prime=-2

integer ideal multiples=[-1, -1];corresponding INPUT cosets agree

representative choice0 addition integer addition := sum(1, 1)

representative choice0 addition integer addition = 2

representative choice0 addition parent mod n exact quotient fraction := div(2, 6)

representative choice0 addition parent mod n exact quotient fraction = 1/3

representative choice0 addition parent mod n:integer value=2;positive modulus=6;exact ratio=1/3

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

representative choice0 addition parent mod n quotient multiple := mul(0, 6)

representative choice0 addition parent mod n quotient multiple = 0

representative choice0 addition parent mod n canonical nonnegative residue := sub(2, 0)

representative choice0 addition parent mod n canonical nonnegative residue = 2

representative choice0 addition quotient class mod d exact quotient fraction := div(2, 2)

representative choice0 addition quotient class mod d exact quotient fraction = 1

representative choice0 addition quotient class mod d:integer value=2;positive modulus=2;exact ratio=1

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

representative choice0 addition quotient class mod d quotient multiple := mul(1, 2)

representative choice0 addition quotient class mod d quotient multiple = 2

representative choice0 addition quotient class mod d canonical nonnegative residue := sub(2, 2)

representative choice0 addition quotient class mod d canonical nonnegative residue = 0

representative choice0 addition:original parent representatives=[1, 1];integer result=2;parent modulus=6

parent result=2;quotient class label=0;class modulus=2

representative choice0 multiplication integer multiplication := mul(1, 1)

representative choice0 multiplication integer multiplication = 1

representative choice0 multiplication parent mod n exact quotient fraction := div(1, 6)

representative choice0 multiplication parent mod n exact quotient fraction = 1/6

representative choice0 multiplication parent mod n:integer value=1;positive modulus=6;exact ratio=1/6

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

representative choice0 multiplication parent mod n quotient multiple := mul(0, 6)

representative choice0 multiplication parent mod n quotient multiple = 0

representative choice0 multiplication parent mod n canonical nonnegative residue := sub(1, 0)

representative choice0 multiplication parent mod n canonical nonnegative residue = 1

representative choice0 multiplication quotient class mod d exact quotient fraction := div(1, 2)

representative choice0 multiplication quotient class mod d exact quotient fraction = 1/2

representative choice0 multiplication quotient class mod d:integer value=1;positive modulus=2;exact ratio=1/2

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

representative choice0 multiplication quotient class mod d quotient multiple := mul(0, 2)

representative choice0 multiplication quotient class mod d quotient multiple = 0

representative choice0 multiplication quotient class mod d canonical nonnegative residue := sub(1, 0)

representative choice0 multiplication quotient class mod d canonical nonnegative residue = 1

representative choice0 multiplication:original parent representatives=[1, 1];integer result=1;parent modulus=6

parent result=1;quotient class label=1;class modulus=2

representative choice1 addition integer addition := sum(3, 3)

representative choice1 addition integer addition = 6

representative choice1 addition parent mod n exact quotient fraction := div(6, 6)

representative choice1 addition parent mod n exact quotient fraction = 1

representative choice1 addition parent mod n:integer value=6;positive modulus=6;exact ratio=1

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

representative choice1 addition parent mod n quotient multiple := mul(1, 6)

representative choice1 addition parent mod n quotient multiple = 6

representative choice1 addition parent mod n canonical nonnegative residue := sub(6, 6)

representative choice1 addition parent mod n canonical nonnegative residue = 0

representative choice1 addition quotient class mod d exact quotient fraction := div(0, 2)

representative choice1 addition quotient class mod d exact quotient fraction = 0

representative choice1 addition quotient class mod d:integer value=0;positive modulus=2;exact ratio=0

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

representative choice1 addition quotient class mod d quotient multiple := mul(0, 2)

representative choice1 addition quotient class mod d quotient multiple = 0

representative choice1 addition quotient class mod d canonical nonnegative residue := sub(0, 0)

representative choice1 addition quotient class mod d canonical nonnegative residue = 0

representative choice1 addition:original parent representatives=[3, 3];integer result=6;parent modulus=6

parent result=0;quotient class label=0;class modulus=2

representative choice1 multiplication integer multiplication := mul(3, 3)

representative choice1 multiplication integer multiplication = 9

representative choice1 multiplication parent mod n exact quotient fraction := div(9, 6)

representative choice1 multiplication parent mod n exact quotient fraction = 3/2

representative choice1 multiplication parent mod n:integer value=9;positive modulus=6;exact ratio=3/2

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

representative choice1 multiplication parent mod n quotient multiple := mul(1, 6)

representative choice1 multiplication parent mod n quotient multiple = 6

representative choice1 multiplication parent mod n canonical nonnegative residue := sub(9, 6)

representative choice1 multiplication parent mod n canonical nonnegative residue = 3

representative choice1 multiplication quotient class mod d exact quotient fraction := div(3, 2)

representative choice1 multiplication quotient class mod d exact quotient fraction = 3/2

representative choice1 multiplication quotient class mod d:integer value=3;positive modulus=2;exact ratio=3/2

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

representative choice1 multiplication quotient class mod d quotient multiple := mul(1, 2)

representative choice1 multiplication quotient class mod d quotient multiple = 2

representative choice1 multiplication quotient class mod d canonical nonnegative residue := sub(3, 2)

representative choice1 multiplication quotient class mod d canonical nonnegative residue = 1

representative choice1 multiplication:original parent representatives=[3, 3];integer result=9;parent modulus=6

parent result=3;quotient class label=1;class modulus=2

ANY changes a-a_prime,b-b_prime in same two-sided ideal I;actual addition classes=[0, 0];actual multiplication classes=[1, 1]

(a+b)-(a_prime+b_prime)=(a-a_prime)+(b-b_prime) in I;ab-a_prime*b_prime=a*(b-b_prime)+(a-a_prime)*b_prime in I;both universal operation obligations=是

逐步反向核對

• S065 · 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。
• S064 · 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。
• S063 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication quotient class mod d canonical nonnegative residue，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication quotient class mod d quotient multiple，不以小數近似替代。
• S061 · 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。
• S060 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication quotient class mod d exact quotient fraction，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication parent mod n canonical nonnegative residue，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication parent mod n quotient multiple，不以小數近似替代。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication parent mod n exact quotient fraction，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 representative choice1 multiplication integer multiplication，不以小數近似替代。
• S054 · 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。
• S053 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition quotient class mod d canonical nonnegative residue，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition quotient class mod d quotient multiple，不以小數近似替代。
• S051 · 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。
• S050 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition quotient class mod d exact quotient fraction，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition parent mod n canonical nonnegative residue，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition parent mod n quotient multiple，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition parent mod n exact quotient fraction，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 representative choice1 addition integer addition，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication quotient class mod d canonical nonnegative residue，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication quotient class mod d quotient multiple，不以小數近似替代。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication quotient class mod d exact quotient fraction，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication parent mod n canonical nonnegative residue，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication parent mod n quotient multiple，不以小數近似替代。
• S037 · 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。
• S036 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication parent mod n exact quotient fraction，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 representative choice0 multiplication integer multiplication，不以小數近似替代。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition quotient class mod d canonical nonnegative residue，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition quotient class mod d quotient multiple，不以小數近似替代。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition quotient class mod d exact quotient fraction，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition parent mod n canonical nonnegative residue，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition parent mod n quotient multiple，不以小數近似替代。
• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition parent mod n exact quotient fraction，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 representative choice0 addition integer addition，不以小數近似替代。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 second representative change divided by d，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 first representative change divided by d，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 second representative change，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 first representative change，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 representative slot3 input class canonical nonnegative residue，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 representative slot3 input class quotient multiple，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 representative slot3 input class exact quotient fraction，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 representative slot2 input class canonical nonnegative residue，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 representative slot2 input class quotient multiple，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 representative slot2 input class exact quotient fraction，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 representative slot1 input class canonical nonnegative residue，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 representative slot1 input class 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 representative slot1 input class exact quotient fraction，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 representative slot0 input class canonical nonnegative residue，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 representative slot0 input class 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 representative slot0 input class 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

([1, 1, 3, 3], [0, 0], [1, 1], 是, 是, 是)

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。