第 15989 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":20,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7}。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":20,"d":5,"a":3,"b":2,"a_prime":8,"b_prime":7}。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(20, 5)

quotient ring parent modulus divided by ideal generator = 4

same declared parent Z_20;generator d=5;n/d=4;ideal multiples=[0, 5, 10, 15]

I=kernel of residue map Z_20 to Z_5;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, 18, 19];same ideal I=[0, 5, 10, 15]

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

representative slot0 input class exact quotient fraction := div(3, 5)

representative slot0 input class exact quotient fraction = 3/5

representative slot0 input class:integer value=3;positive modulus=5;exact ratio=3/5

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

representative slot0 input class quotient multiple := mul(0, 5)

representative slot0 input class quotient multiple = 0

representative slot0 input class canonical nonnegative residue := sub(3, 0)

representative slot0 input class canonical nonnegative residue = 3

representative slot1 input class exact quotient fraction := div(2, 5)

representative slot1 input class exact quotient fraction = 2/5

representative slot1 input class:integer value=2;positive modulus=5;exact ratio=2/5

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

representative slot1 input class quotient multiple := mul(0, 5)

representative slot1 input class quotient multiple = 0

representative slot1 input class canonical nonnegative residue := sub(2, 0)

representative slot1 input class canonical nonnegative residue = 2

representative slot2 input class exact quotient fraction := div(8, 5)

representative slot2 input class exact quotient fraction = 8/5

representative slot2 input class:integer value=8;positive modulus=5;exact ratio=8/5

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

representative slot2 input class quotient multiple := mul(1, 5)

representative slot2 input class quotient multiple = 5

representative slot2 input class canonical nonnegative residue := sub(8, 5)

representative slot2 input class canonical nonnegative residue = 3

representative slot3 input class exact quotient fraction := div(7, 5)

representative slot3 input class exact quotient fraction = 7/5

representative slot3 input class:integer value=7;positive modulus=5;exact ratio=7/5

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

representative slot3 input class quotient multiple := mul(1, 5)

representative slot3 input class quotient multiple = 5

representative slot3 input class canonical nonnegative residue := sub(7, 5)

representative slot3 input class canonical nonnegative residue = 2

first representative change := sub(3, 8)

first representative change = -5

second representative change := sub(2, 7)

second representative change = -5

first representative change divided by d := div(-5, 5)

first representative change divided by d = -1

second representative change divided by d := div(-5, 5)

second representative change divided by d = -1

original representative choices=[3, 2, 8, 7];class labels in slot order=[3, 2, 3, 2];changes a-a_prime=-5;b-b_prime=-5

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

representative choice0 addition integer addition := sum(3, 2)

representative choice0 addition integer addition = 5

representative choice0 addition parent mod n exact quotient fraction := div(5, 20)

representative choice0 addition parent mod n exact quotient fraction = 1/4

representative choice0 addition parent mod n:integer value=5;positive modulus=20;exact ratio=1/4

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

representative choice0 addition parent mod n quotient multiple := mul(0, 20)

representative choice0 addition parent mod n quotient multiple = 0

representative choice0 addition parent mod n canonical nonnegative residue := sub(5, 0)

representative choice0 addition parent mod n canonical nonnegative residue = 5

representative choice0 addition quotient class mod d exact quotient fraction := div(5, 5)

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

representative choice0 addition quotient class mod d:integer value=5;positive modulus=5;exact ratio=1

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

representative choice0 addition quotient class mod d quotient multiple := mul(1, 5)

representative choice0 addition quotient class mod d quotient multiple = 5

representative choice0 addition quotient class mod d canonical nonnegative residue := sub(5, 5)

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

representative choice0 addition:original parent representatives=[3, 2];integer result=5;parent modulus=20

parent result=5;quotient class label=0;class modulus=5

representative choice0 multiplication integer multiplication := mul(3, 2)

representative choice0 multiplication integer multiplication = 6

representative choice0 multiplication parent mod n exact quotient fraction := div(6, 20)

representative choice0 multiplication parent mod n exact quotient fraction = 3/10

representative choice0 multiplication parent mod n:integer value=6;positive modulus=20;exact ratio=3/10

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

representative choice0 multiplication parent mod n quotient multiple := mul(0, 20)

representative choice0 multiplication parent mod n quotient multiple = 0

representative choice0 multiplication parent mod n canonical nonnegative residue := sub(6, 0)

representative choice0 multiplication parent mod n canonical nonnegative residue = 6

representative choice0 multiplication quotient class mod d exact quotient fraction := div(6, 5)

representative choice0 multiplication quotient class mod d exact quotient fraction = 6/5

representative choice0 multiplication quotient class mod d:integer value=6;positive modulus=5;exact ratio=6/5

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

representative choice0 multiplication quotient class mod d quotient multiple := mul(1, 5)

representative choice0 multiplication quotient class mod d quotient multiple = 5

representative choice0 multiplication quotient class mod d canonical nonnegative residue := sub(6, 5)

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

representative choice0 multiplication:original parent representatives=[3, 2];integer result=6;parent modulus=20

parent result=6;quotient class label=1;class modulus=5

representative choice1 addition integer addition := sum(8, 7)

representative choice1 addition integer addition = 15

representative choice1 addition parent mod n exact quotient fraction := div(15, 20)

representative choice1 addition parent mod n exact quotient fraction = 3/4

representative choice1 addition parent mod n:integer value=15;positive modulus=20;exact ratio=3/4

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

representative choice1 addition parent mod n quotient multiple := mul(0, 20)

representative choice1 addition parent mod n quotient multiple = 0

representative choice1 addition parent mod n canonical nonnegative residue := sub(15, 0)

representative choice1 addition parent mod n canonical nonnegative residue = 15

representative choice1 addition quotient class mod d exact quotient fraction := div(15, 5)

representative choice1 addition quotient class mod d exact quotient fraction = 3

representative choice1 addition quotient class mod d:integer value=15;positive modulus=5;exact ratio=3

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

representative choice1 addition quotient class mod d quotient multiple := mul(3, 5)

representative choice1 addition quotient class mod d quotient multiple = 15

representative choice1 addition quotient class mod d canonical nonnegative residue := sub(15, 15)

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

representative choice1 addition:original parent representatives=[8, 7];integer result=15;parent modulus=20

parent result=15;quotient class label=0;class modulus=5

representative choice1 multiplication integer multiplication := mul(8, 7)

representative choice1 multiplication integer multiplication = 56

representative choice1 multiplication parent mod n exact quotient fraction := div(56, 20)

representative choice1 multiplication parent mod n exact quotient fraction = 14/5

representative choice1 multiplication parent mod n:integer value=56;positive modulus=20;exact ratio=14/5

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

representative choice1 multiplication parent mod n quotient multiple := mul(2, 20)

representative choice1 multiplication parent mod n quotient multiple = 40

representative choice1 multiplication parent mod n canonical nonnegative residue := sub(56, 40)

representative choice1 multiplication parent mod n canonical nonnegative residue = 16

representative choice1 multiplication quotient class mod d exact quotient fraction := div(16, 5)

representative choice1 multiplication quotient class mod d exact quotient fraction = 16/5

representative choice1 multiplication quotient class mod d:integer value=16;positive modulus=5;exact ratio=16/5

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

representative choice1 multiplication quotient class mod d quotient multiple := mul(3, 5)

representative choice1 multiplication quotient class mod d quotient multiple = 15

representative choice1 multiplication quotient class mod d canonical nonnegative residue := sub(16, 15)

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

representative choice1 multiplication:original parent representatives=[8, 7];integer result=56;parent modulus=20

parent result=16;quotient class label=1;class modulus=5

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

([3, 2, 8, 7], [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。