❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖

❚❘×❒◆● ❚❍➚ ▲■➊◆

▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍⑨ ◆❐■✱ ✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖

❚❘×❒◆● ❚❍➚ ▲■➊◆

▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❑■➋❯ ◆●❆

❍⑨ ◆❐■✱ ✷✵✶✾

▲❮■ ❈❆▼ ✣❖❆◆
❑❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ❝õ❛ ❜↔♥ t❤➙♥ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤
❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❡♠ ❝á♥
✤÷ñ❝ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ✈➔ ♥❤➟♥ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❝æ
tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ sü q✉❛♥ t➙♠✱ ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛
❚❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛✳
❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ❝â t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐
❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❱➻ ✈➟② ❡♠ ①✐♥ ❦❤➥♥❣ ✤à♥❤
✤➲ t➔✐ ✧▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✧ ❦❤æ♥❣ ❝â sü trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲
t➔✐ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝✳

❍➔ ◆ë✐✱ ✻ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥
❚r÷ì♥❣ ❚❤à ▲✐➯♥

✶


▲❮■ ❈❷▼ ❒◆
❑❤â❛ ❧✉➟♥ ♥➔② ✤÷ñ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷
♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ ❚❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉
◆❣❛✳
❊♠ ①✐♥ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tî✐ ❚❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛✱
♥❣÷í✐ ✤➣ ✤à♥❤ ❤÷î♥❣ ✈➔ ❝❤➾ ❞➝♥ s→t s❛♦ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱
♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❙ü ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ♥❣❤✐➯♠
tó❝ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥❤ú♥❣ ✤à♥❤ ❤÷î♥❣ ✤ó♥❣ ✤➢♥ ❝õ❛ ❝æ ❧➔ t✐➲♥ ✤➲
q✉❛♥ trå♥❣ ❣✐ó♣ ❡♠ ❝â ✤÷ñ❝ ♥❤ú♥❣ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥
♥➔②✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ❝→❝ t❤➛②
❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ tr♦♥❣ ❦❤♦❛ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ❡♠

❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tî✐ ❜è ♠➭ ✈➔ ❣✐❛ ✤➻♥❤ ❝õ❛ ❡♠✱
♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❝❤✐❛ s➫ ✈➔ ✤ë♥❣ ✈✐➯♥ ❡♠ ✤➸ ❡♠ ❝è ❣➢♥❣ ❤♦➔♥ t❤➔♥❤
❦❤â❛ ❧✉➟♥ ♥➔②✳

❍➔ ◆ë✐✱ ✻ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥
❚r÷ì♥❣ ❚❤à ▲✐➯♥

✷


▼Ö❈ ▲Ö❈
❚r❛♥❣

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳

✶
✷

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳

✹

▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
❑➼ ❤✐➺✉

▲❮■ ▼Ð ✣❺❯ ✳


✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳

✻

✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳ ▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳ ▼æ✤✉♥ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳ ▼æ✤✉♥ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✶✳ ❳➙② ❞ü♥❣ ♠æ✤✉♥ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷✳ ▼ët sè ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹✳ ❚➼❝❤ trü❝ t✐➳♣✱ tê♥❣ trü❝ t✐➳♣ ✈➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛
♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺✳ ✣ç♥❣ ❝➜✉ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳ ❉➣② ❦❤î♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽
✽
✶✵
✶✵
✶✵
✶✶
✶✶
✶✷
✶✻

✷✳ ▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳ ❚➟♣ s✐♥❤ ❝õ❛ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳ ▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶✳ ▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥


✷✵
✷✵
✷✵
✷✷

✸

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳


✷✳✷✳✷✳ ▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✸✳ ▼æ✤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹


❑➑ ❍■➏❯
∅
N
Z
Q
R
C
A
(A, M)
J (A)

❚➟♣ ❤ñ♣ ré♥❣✳
❚➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥✳
❚➟♣ ❝→❝ sè ♥❣✉②➯♥✳
❚➟♣ ❝→❝ sè ❤ú✉ t➾✳
❚➟♣ ❝→❝ sè t❤ü❝✳

❚➟♣ ❝→❝ sè ♣❤ù❝✳
❱➔♥❤✳
❱➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈î✐ ✐✤➯❛♥ ❝ü❝ ✤↕✐ M✳
❈➠♥ ❏❛❝♦❜s♦♥ ❝õ❛ ✈➔♥❤ ❆✳

✺


é
ỵ ồ t
ồ t tr trú số
õ ởt t ợ sỹ t tr õ r trú số t
trú ổ t t ỵ tt t ồ
õ ỡ s t tr ỵ tt trú ổ ữủ
ữ ổ ỳ s ổ tỹ ổ ở
ổ tr ổ rt r õ ổ ỳ s ởt
ổ õ trỏ q trồ tr ỵ tt ổ t s
ỡ trú ổ tổ ỹ ồ t ổ ỳ
s õ tốt ố õ ừ

ố tữủ ự
ự ổ ỳ s

ử ở ự
ử ừ õ tự ừ t
õ tự s s ỡ tự ổ ỳ s
ở õ ỗ ữỡ


ữỡ tự

r ữỡ ú tổ ổ
ổ ổ tữỡ t trỹ t tờ trỹ t tỷ
trỹ t ỗ ổ ợ tữỡ ữỡ



❝õ❛ ❞➣② ❦❤î♣ ♥❣➢♥✳
•

❈❤÷ì♥❣ ✷✿ ▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➟♣ s✐♥❤ ❝õ❛ ♠æ✤✉♥✱
✤à♥❤ ♥❣❤➽❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱ ❝→❝ ✤à♥❤ ❧➼✱ ❤➺ q✉↔ ✈➔ ❜ê ✤➲ ❧✐➯♥
q✉❛♥ ✤➳♥ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ♠æ✤✉♥ ❤ú✉
❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ ♠æ✤✉♥ ◆♦❡t❤❡r ✈➔ ♠ët sè ❜➔✐ t➟♣✳

❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ♥➠♥❣ ❧ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥
❦❤â❛ ❧✉➟♥ ❦❤â tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât✳ ❊♠ r➜t ♠æ♥❣ ♥❤➟♥ ✤÷ñ❝ sü ✤â♥❣
❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷ñ❝
❤♦➔♥ t❤✐➺♥ ❤ì♥✳

❍➔ ◆ë✐✱ ✻ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥
❚r÷ì♥❣ ❚❤à ▲✐➯♥

✼


❈❤÷ì♥❣ ✶

❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♠æ✤✉♥
♥❤➡♠ sû ❞ö♥❣ ❝❤♦ ❝❤÷ì♥❣ s❛✉✳

✶✳✶✳ ▼æ✤✉♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ❝â ✤ì♥ ✈à ✶✳ ▼ët ♠æ✤✉♥ tr→✐

tr➯♥ A ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝ë♥❣ M ❝ò♥❣ ✈î✐ ♣❤➨♣ ♥❤➙♥ ✈æ ❤÷î♥❣ ❝→❝
♣❤➛♥ tû ❝õ❛ A✳
•:A×M →M

t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✐✮ r(m + m ) = rm + rm ;
✐✐✮ (r + r )m = rm + r m;
✐✐✐✮ (rr )m = r(r m);
✐✈✮ 1.m = m;
✈î✐ ♠å✐ r, r

∈ A, m, m ∈ M.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ▼æ✤✉♥

♣❤↔✐ tr➯♥ A ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝ë♥❣ M
❝ò♥❣ ✈î✐ ♣❤➨♣ ♥❤➙♥ ✈æ ❤÷î♥❣ ❝→❝ ♣❤➛♥ tû ❝õ❛ A✿
•:M ×A→M

t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✽


✐✮ (m + m )r = rm + rm ;

✐✐✮ m(r + r ) = rm + r m;
✐✐✐✮ m(rr ) = (mr)r ;
✐✈✮ m.1 = m;
✈î✐ ♠å✐ r, r

∈ A, m, m ∈ M.

◆❤➟♥ ①➨t ✶✳✶✳✶✳ ◆➳✉ A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ t❤➻ ♠æ✤✉♥ tr→✐ trò♥❣ ✈î✐

♠æ✤✉♥ ♣❤↔✐✳ ❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ t❛ s➩ ❝❤➾ ①➨t ❝→❝ A ✲ ♠æ✤✉♥ tr→✐ ✈➔
❣å✐ t➢t ❧➔ A ✲ ♠æ✤✉♥✳ ◆➳✉ A ❧➔ ♠ët tr÷í♥❣ t❤➻ A ✲ ♠æ✤✉♥ ❣å✐ ❧➔ ♠ët
❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ A ❤❛② A ✲ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì✳
❱➼ ❞ö ✶✳✶✳✶✳ ▼å✐ ♥❤â♠ ❆❜❡❧ ❝ë♥❣ M ❧➔ Z ✲ ♠æ✤✉♥✳
❚❤➟t ✈➟②✱ ✈î✐ ♠å✐ n ∈ Z✱ x ∈ M ✳ ✣➦t✿

nx =




x + x + ... + x





 n sè ❤↕♥❣
0







(−x) + (−x) + ... + (−x)




♥➳✉ ♥❃✵
♥➳✉ ♥❂✵
♥➳✉ ♥❁✵

|n| sè ❤↕♥❣

❉♦ ✤â nx ∈ M ✳ ❙✉② r❛ tç♥ t↕✐ →♥❤ ①↕
Z×M →M
(n, x) → nx

✈➔ ♣❤➨♣ t♦→♥ tr➯♥ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ t➼❝❤ ✈æ ❤÷î♥❣✳
❱➼ ❞ö ✶✳✶✳✷✳ ◆➳✉ A ❧➔ ✈➔♥❤ ❝â ✤ì♥ ✈à t❤➻ (A, +) ❧➔ ♥❤â♠ ❆❜❡❧ ✈➔
♣❤➨♣ ♥❤➙♥ tr➯♥ ✈➔♥❤ A t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ t➼❝❤ ✈æ ❤÷î♥❣✳ ❱➻
t❤➳ A ❧➔ A ✲ ♠æ✤✉♥✳
✾


✶✳✷✳ ▼æ✤✉♥ ❝♦♥
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ M

❧➔ ♠ët ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ A✱

G ⊂ M ✳ ❚❛ ♥â✐ G ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❤♦➦❝ ♠ët A ✲ ♠æ✤✉♥ ❝♦♥
❝õ❛ M ♥➳✉ G ❝ò♥❣ ❤❛✐ ♣❤➨♣ t♦→♥ ❝↔♠ s✐♥❤ tr➯♥ M ❧➔ A ✲ ♠æ✤✉♥✳
✣✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ ✶✳✷✳✶✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ M
❧➔ A ✲ ♠æ✤✉♥ ✈➔ G ⊂ M, G = ∅✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ G ❧➔ ♠æ✲✤✉♥ ❝♦♥ ❝õ❛ M ;
✭✐✐✮ ❱î✐ ♠å✐ g, g

∈ G, r ∈ A

t❤➻ g + g

∈G

✈➔ rg ∈ G;

✭✐✐✐✮ ❱î✐ ♠å✐ g, g ∈ G ✈➔ ♠å✐ r, r ∈ A t❤➻ rg + r g ∈ G.
❱➼ ❞ö ✶✳✷✳✶✳ ●✐↔ sû M ❧➔ A ✲ ♠æ✤✉♥✱ ■ ❧➔ ✐✤➯❛♥ ❝õ❛ A✱
IM =
i∈I αi xi | αi ∈ I, xi ∈ M ❧➔ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳
❱➼ ❞ö ✶✳✷✳✷✳ ❈❤♦ M ❧➔ A ✲ ♠æ✤✉♥✳ ❚❛ ❝â {0} ✈➔ M ❧➔ A ✲ ♠æ✤✉♥ ❝♦♥
❝õ❛ M ❣å✐ ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ t➛♠ t❤÷í♥❣ ❝õ❛ M ✳

✶✳✸✳ ▼æ✤✉♥ t❤÷ì♥❣
✶✳✸✳✶✳ ❳➙② ❞ü♥❣ ♠æ✤✉♥ t❤÷ì♥❣
❈❤♦ M ❧➔ ♠ët ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ A ✈➔ G ❧➔ A ✲ ♠æ✤✉♥
❝♦♥ ❝õ❛ M ✳ ◆➯♥ G ❧➔ ♥❤â♠ ❝ë♥❣ ❆❜❡❧ ❝õ❛ M ✱ ❞♦ ✤â G ❧➔ ♥❤â♠ ❝♦♥
❝❤✉➞♥ t➢❝ ❝õ❛ M ✳ ❑❤✐ ✤â tç♥ t↕✐ ♥❤â♠ t❤÷ì♥❣ M/G ❧➔ ♥❤â♠ ❝ë♥❣ ❆❜❡❧
✈î✐ ♣❤➨♣ ❝ë♥❣ ✿
x + G + y + G = x + y + G.


❚r➯♥ M/G ①→❝ ✤à♥❤ ♣❤➨♣ ♥❤➙♥ ✈æ ❤÷î♥❣ ✈î✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ A ♥❤÷
s❛✉✿
❱î✐ ♠å✐ α ∈ A ✱ x + G ∈ M/G t❤➻ α(x + G) = αx + G.
❙✉② r❛ M/G ❧➔ A ✲ ♠æ✤✉♥ t❤÷ì♥❣ ❝õ❛ M tr➯♥ ♠æ✤✉♥ ❝♦♥ G ❝õ❛ ♥â✳
✶✵


ởt số ử
ử M

ởt A ổ M {0} ổ
ừ M õ t õ ổ tữỡ
M/ = {x + M | x M } = {M } .
M
M/
{0} = {x + {0} |x M } = {x | x M } = M.

ử õ ở R Z ổ õ ở Q Z ổ

ừ R õ tỗ t ổ tữỡ R/Q
R/Q = {x + Q | x R} õ x Q t x + Q = Q
R/Q = Q, x + Q ợ x R/Q ợ ở ổ
ữợ
(x + Q) + (y + Q) = x + y + Q.
(x + Q) = x + Q.

trỹ t tờ trỹ t tỷ trỹ
t ừ ổ
trỹ t {Mi}iI


ởt ồ tũ ỵ
A ổ r t iI Mi t t
ở ợ ổ ữợ ữ s
ợ a R, (xi)iI , (yi)iI iI Mi
(xi )iI + (yi )iI = (xi + yi )iI .
a (xi )iI = (axi )iI .

t ữủ iI Mi ũ ợ t õ tr ởt
A ổ ồ t trỹ t ừ ồ ổ {Mi }iI .



ờ trỹ t {Mi}iI

ởt ồ
A ổ (xi )iI , (xi ) Mi ồ õ ỳ xi = 0
t trứ r ởt số ỳ số ồ iI Mi t ủ ỗ
(xi )iI õ ỳ õ
iI Mi ũ ợ t ở
ổ ữợ ởt A ổ ồ iI Mi tờ trỹ t
ừ ồ ổ {Mi}iI .
t ờ trỹ t ổ ừ t trỹ t
I t số ỳ tự I = {1, 2, ..., n} t tờ trỹ t
t trỹ t trũ
tỷ trỹ t
N ởt A ổ ừ A ổ M õ r N
ởt tỷ trỹ t ừ M tỗ t ởt A ổ P ừ
M s M = N P õ t õ r P ổ ử ừ
N tr M
ổ M = 0 ữủ ồ ổ t ữủ M

ỳ tỷ trỹ t t tr M
ử sỷ V ởt K ổ tỡ {ei|i I}
ởt ỡ s ừ V õ V = iI eiK.
Z Z ổ õ Z ổ t ữủ t ồ
ổ tỹ sỹ ừ Z õ mZ, m = 0, 1. mZ ởt
tỷ trỹ t ừ Z t tỗ t ởt ổ nZ ợ n = 0, 1
s Z = mZ nZ. õ mn mZ nZ = 0 r m = 0
n = 0 t

ỗ ổ
M, N

A ổ f : M N
ữủ ồ ởt ỗ A ổ ỏ ồ A ỗ



✈î✐ ♠å✐ x, y ∈ M ✈➔ ♠å✐ a ∈ R t❤➻✿
✐✮ f (x + y) = x + y;
✐✐✮ f (ax) = af (x) .
❈❤ó þ ✶✳✺✳✶✳ ❚❛

♥â✐ ❤❛✐ ♠æ✤✉♥ M ✈➔ N ✤➥♥❣ ❝➜✉ ✈î✐ ♥❤❛✉✱ ✈✐➳t
M N ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ ✤➥♥❣ ❝➜✉ ♠æ✤✉♥ f : M → N ✳
❱➼ ❞ö ✶✳✺✳✶✳ ✭✶✮ ❈❤♦ M, N ❧➔ ❝→❝ A ✲ ♠æ✤✉♥✿
Θ: M →N
x→0

❧➔ ❝→❝ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳
✭✷✮ ❈❤♦ N ❧➔ ❝→❝ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ⑩♥❤ ①↕✿

p : M → M/N
x→x+N

❧➔ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳ ❍ì♥ ♥ú❛ p ❧➔ t♦➔♥ ❝➜✉ ✈➔ ❣å✐ ❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤
t➢❝✳ ❚❤➟t ✈➟②✱ ✈î✐ ♠å✐ x, y ∈ M ✱ α ∈ R t❛ ❝â✿
p(x + y) = x + y + N = (x + N ) + (y + N ) = p (x) + p (y) .
p (αx) = αx + N = α (x + N ) = αp (x) .

❙✉② r❛ ♣ ❧➔ A ✲ ✤ç♥❣ ❝➜✉✳
❱î✐ ♠å✐ y ∈ M/N t❤➻ tç♥ t↕✐ x ∈ M ✤➸ y = x + N ❤❛② tç♥ t↕✐ x ∈ N
✤➸ p (x) = x + N = y✳ ❙✉② r❛ p ❧➔ t♦➔♥ →♥❤✳ ❱➟② p ❧➔ t♦➔♥ ❝➜✉✳
✭✸✮ ❈❤♦ {Mi}i∈I ❤å ❝→❝ A ✲ ♠æ✤✉♥✱ i∈I Mi ❧➔ t➼❝❤ trü❝ t✐➳♣ ❤å ❝→❝
♠æ✤✉♥ {Mi}i∈I ✳ ❈→❝ →♥❤ ①↕✿
Mi → Mi

pi :
i∈I

✶✸


(xi )i∈I → xi

✈➔
qi : Mi →

Mi
i∈I

xi → (0, ..., 0, xi , 0, ..., 0)


❙✉② r❛ pi, qi ❧➔ ❝→❝ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳
✣à♥❤ ❧➼ ✶✳✺✳✶✳ ✭✣✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣✮✳ ⑩♥❤ ①↕ f : M → N ❧➔ A ✲
✤ç♥❣ ❝➜✉ ♠æ✤✉♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
f (αx + βy) = αf (x) + βf (y)

✈î✐ ♠å✐ α, β ∈ A ✈➔ ♠å✐ x, y ∈ M ✳
◆❤➟♥ ①➨t ✶✳✺✳✶✳ ❈❤♦ f : M → N ❧➔ A ✲ ✤ç♥❣ ❝➜✉✳ ❙✉② r❛ f ❧➔ ✤ç♥❣
❝➜✉ ♥❤â♠ ✤è✐ ✈î✐ ♣❤➨♣ t♦→♥ ❝ë♥❣✱ s✉② r❛ f ❝â t➜t ❝↔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛
✤ç♥❣ ❝➜✉ ♥❤â♠ ✤è✐ ✈î✐ ♣❤➨♣ t♦→♥ ❝ë♥❣✳
◆❣♦➔✐ r❛ t❛ ❝â ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥ s❛✉✿
❛✮ ❈❤♦ f : M → N s✉② r❛ N ❧➔ A ✲ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳ A, B ❧➛♥ ❧÷ñt ❧➔
❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✈➔ N ✳ ❑❤✐ ✤â f (A) , f −1 (B) ❧➛♥ ❧÷ñt ❧➔ ❝→❝ A ✲
♠æ✤✉♥ ❝♦♥ ❝õ❛ N ✈➔ M.
f (A) = {f (x) | x ∈ A} .
f −1 (B) = {x ∈ M | f (x) ∈ N } .

✣➦❝ ❜✐➺t✱ ❝❤♦ f : M → N ❧➔ A ✲ ♠æ✤✉♥✳
Kerf = {x ∈ M | f (x) = 0} = f −1 (0) ❣å✐ ❧➔ ❤↕t ♥❤➙♥ ❝õ❛ X ✳
Imf = f (M ) = {f (x) | x ∈ M } ❣å✐ ❧➔ ↔♥❤ ❝õ❛ M ✳
❚❛ ❝â M ✈➔ {0} ❧➛♥ ❧÷ñt ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✈➔ N ✳ ❙✉② r❛ Imf
✈➔ Kerf ❧➛♥ ❧÷ñt ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ N ✈➔ M ✳
✶✹


❜✮ ✣à♥❤ ❧➼ tê♥❣ q✉→t ✤ç♥❣ ❝➜✉ ♠æ✤✉♥
❈❤♦ f : M → N ❧➔ A ✲ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳ E, F ❧➛♥ ❧÷ñt ❧➔ ❝→❝ ♠æ✤✉♥
❝♦♥ ❝õ❛ M ✈➔ N s❛♦ ❝❤♦ f (E) ⊂ F ✱ pE : M → M/E ✱ pF : N → N/F
❧➔ ❝→❝ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❑❤✐ ✤â tç♥ t↕✐ ❞✉② ♥❤➜t A ✲ ✤ç♥❣ ❝➜✉
♠æ✤✉♥ f : M/E → N/F s❛♦ ❝❤♦ f .pE = pF .f ✱ tù❝ ❧➔ ❜✐➸✉ ✤ç s❛✉ ❣✐❛♦

❤♦→♥✿

❝✮ ❍➺ q✉↔

❈❤♦ f : M → N ❧➔ ❝→❝ A ✲ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✱ B = Kerf ✱ pB : M →
M/ ❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❑❤✐ ✤â tç♥ t↕✐ A ✲ ✤ç♥❣ ❝➜✉ f : M/ → N
B
B
s❛♦ ❝❤♦ f .pB = f ✱ tù❝ ❧➔ Imf = Imf ✈➔ ❜✐➸✉ ✤ç s❛✉ ❣✐❛♦ ❤♦→♥✿

❞✮ ❍➺ q✉↔

❈❤♦ f : M → N ❧➔ ❝→❝ A ✲ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✳ ❑❤✐ ✤â✿
✶✺


✭✶✮ M/Kerf ∼= Imf ;
✭✷✮ ◆➳✉ f ❧➔ t♦➔♥ ❝➜✉ t❤➻ M/Kerf ∼= N ✳
❡✮ ❍➺ q✉↔
❈❤♦ B, C ❧➔ ❝→❝ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ A ✲ ♠æ✤✉♥ ▼✳ ❑❤✐ ✤â
B+C ∼ B
=
C
B∩C

❢✮ ❍➺ q✉↔
❈❤♦ H ❧➔ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ I ✱ I ❧➔ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ K ✳ ❚❛ ❝â
K/
K/ ∼
H

=
I

I/ .
H

❣✮ ❈❤♦ f : M → N ❧➔ ❝→❝ A ✲ ✤ç♥❣ ❝➜✉✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②
t÷ì♥❣ ✤÷ì♥❣✿
✭✶✮ f ❧➔ A ✲ ✤ç♥❣ ❝➜✉ t➛♠ t❤÷í♥❣ ✭tù❝ ❧➔ f = Θ✮❀
✭✷✮ Imf = {0} ;
✭✸✮ Kerf = N ✳
❤✮ ❈❤♦ f : L → M ✈➔ ❈❤♦ g : M → K ✳ ❑❤✐ ✤â h = gof ❧➔ A ✲
✤ç♥❣ ❝➜✉ t➛♠ t❤÷í♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Imf ⊂ Kerf ✳

✶✳✻✳ ❉➣② ❦❤î♣
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✶✳ ✭✐✮

❈❤♦ i ❧➔ sè tü ♥❤✐➯♥✱ ❞➣② ❝→❝ A ✲ ♠æ✤✉♥ ✈➔
❝→❝ A ✲ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥✿
f
f
f
✭✶✳✶✮
... → Mi+1 → Mi → Mi−1 → ...
●å✐ ❧➔ ❞➣② ❦❤î♣ t↕✐ i ♥➳✉ Imfi+1 = Kerfi✳
✭✐✐✮ ❉➣② ✭✶✳✶✮ ❣å✐ ❧➔ ❞➣② ❦❤î♣ ♥➳✉ ♥â ❦❤î♣ t↕✐ i✱ ✈î✐ ♠å✐ i ∈ I ✳
i+1

i


✶✻

i−1


▼➺♥❤ ✤➲ ✶✳✻✳✶✳

❉➣② ❦❤î♣ 0 →k M

✭✣✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❞➣② ❦❤î♣ ♥❣➢♥✮✳
f
g
h
→ M → M → 0 ❧➔ ❞➣② ❦❤î♣ ♥❣➢♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐



❢




❧➔ ✤ì♥ ❝➜✉
❣ ❧➔ t♦➔♥ ❝➜✉




■♠❢❂❑❡r❢
❱➼ ❞ö ✶✳✻✳✶✳ ●✐↔ sû G ❧➔ A ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✱ i ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝❤➼♥❤


t➢❝ ✈➔ p ❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❚❛ ❝â ❞➣② ❦❤î♣ ♥❣➢♥✿
i

p

0 → G → M → M/G → 0.

❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
→ M → M → 0 ❧➔ ❞➣② ❦❤î♣ ♥❣➢♥❀

▼➺♥❤ ✤➲ ✶✳✻✳✷✳

✐✮ 0 → M

✐✐✮ ❚ç♥ t↕✐ ♠æ✤✉♥ ❝♦♥ L ❝õ❛ M s❛♦ ❝❤♦ M ∼= L ✈➔ M ∼= M/L✳
f
g
h
✣à♥❤ ❧➼ ✶✳✻✳✸✳ ❚r♦♥❣ ♠ët s➣② ❦❤î♣ tò② þ B → C → D →
E ❝õ❛ ❝→❝
A ✲ ♠æ✤✉♥ ✈➔ ❝→❝ ✤ç♥❣ ❝➜✉✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
✐✮ f ❧➔ t♦➔♥ ❝➜✉❀
✐✐✮ g ❧➔ ✤ç♥❣ ❝➜✉ t➛♠ t❤÷í♥❣❀
✐✐✐✮ h ❧➔ ✤ì♥ ❝➜✉✳
❍➺ q✉↔ ✶✳✻✳✹✳ ❈❤♦ ♠ët ❞➣② ❦❤î♣ tò② þ ❝→❝ A ✲ ♠æ✤✉♥ ✈➔ ❝→❝ A ✲
✤ç♥❣ ❝➜✉✿ B →f C →g D →h E →k G. ❑❤✐ ✤â C = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿

❢


❧➔ t♦➔♥ ❝➜✉
❦ ❧➔ ✤ì♥ ❝➜✉
◆➳✉ ❞➣② ❝→❝ A ✲ ✤ç♥❣ ❝➜✉ ✈➔ ❝→❝ A ✲ ♠æ✤✉♥
0 → C → 0 ❧➔ ❦❤î♣ t❤➻ C = 0.

❍➺ q✉↔ ✶✳✻✳✺✳

✶✼


A ỗ A ổ
f
0 C D 0 ợ t f
q ởt ợ tũ ỵ A ổ A
ỗ B d C f D g E h F k G.
s tữỡ ữỡ

q

g
f h ỗ t tữớ
d t k ỡ
ợ r

ợ

... X Y Z ...




ồ r t ổ Y = Imf B Imf
ởt tỷ trỹ t ừ
ồ r õ r t ồ ổ ổ
ừ õ
t t ợ
f

k

g

h

0BCD0
B = 0 B, Imk = 0, B = Imk B

t B



s r ợ r

s r D = Img 0 r t
C ợ r r t C

D = D 0, Imf = Kerh = D

ử M, N

A ổ t

pN

i

M
0M
M N N 0




✈î✐ iM (x) = x0, pN (x, y) = y t❤➻ ❞➣② tr➯♥ ❧➔ ❞➣② ❦❤î♣ ♥❣➢♥✳
❚❤➟t ✈➟②✱ Imi = i (M ) = {i (x) | ∀x ∈ M } = {(x, 0) | ∀x ∈ M } ∼
= M✳
❈❤♦ ♥➯♥ ❞➣② tr➯♥ ❝❤➫ r❛ t↕✐ M ⊕ N ✳
✣à♥❤ ❧➼ ✶✳✻✳✽✳ ◆➳✉ ♠ët ❞➣② ❦❤î♣ ... → X → Y → Z → ... ❝❤➫ r❛
t↕✐ ♠æ✤✉♥ Y t❤➻ Y ∼= Imf ⊕ Img✳

✶✾


❈❤÷ì♥❣ ✷

▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ ✈➲ ♠æ✤✉♥ ❤ú✉
❤↕♥ s✐♥❤✱ ❝ö t❤➸ ❧➔ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱
♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥
✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ ♠ët tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ❝õ❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤
❧➔ ♠æ✤✉♥ ◆♦❡t❤❡r✳


✷✳✶✳ ❚➟♣ s✐♥❤ ❝õ❛ ♠æ✤✉♥
✐✮ ●✐↔ sû M ❧➔ A ✲ ♠æ✤✉♥✱ S ∈ M ✳ ●✐❛♦ ❝õ❛
t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❝❤ù❛ S ✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛
M s✐♥❤ ❜ð✐ S ✱ ❦➼ ❤✐➺✉ ❧➔ S ✳ ❚❛ ♥â✐ S ❧➔ t➟♣ s✐♥❤ ❝õ❛ S ❤❛② S
s✐♥❤ r❛ S ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳

✐✐✮ ◆➳✉ S ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ ♠æ✤✉♥ M s❛♦ ❝❤♦ ✈î✐ ♠å✐ t➟♣ ❝♦♥ S S
t❛ ✤➲✉ ❝â S = M t❤➻ S ✤÷ñ❝ ❣å✐ ❧➔ ❤➺ s✐♥❤ ❝ü❝ t✐➸✉ ❝õ❛ ♠æ✤✉♥
M✳
◆❤➟♥ ①➨t ✷✳✶✳✶✳ S ❝❤➼♥❤ ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❤ä ♥❤➜t ❝õ❛ ♠æ✤✉♥ M
❝❤ù❛ S ✳ ◆➳✉ S = M t❤➻ S ❧➔ t➟♣ s✐♥❤ ❝õ❛ M ✳

✷✳✷✳ ▼æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ●✐↔

sû S ❧➔ t➟♣ s✐♥❤ ❝õ❛ M ✳ ◆➳✉ S ❤ú✉ ❤↕♥ t❛
♥â✐ r➡♥❣ M ❧➔ A ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ M ❧➔ ❤ú✉ ❤↕♥
s✐♥❤ ♥➳✉ ❝â ❝→❝ ♣❤➛♥ tû x1, ..., xn ∈ M s❛♦ ❝❤♦ M = x1, ..., xn .
✷✵


◆➳✉ M =

t❤➻ M ✤÷ñ❝ ❣å✐ ❧➔ A ✲ ♠æ✤✉♥ ①②❝❧✐❝✳
❈❤ó þ ✷✳✷✳✶✳ ✭✶✮ S ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❝❤ù❛
S✳
a


✭✷✮ ▼é✐ ♣❤➛♥ tû ❝õ❛

S

❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ tr➯♥ A ❝õ❛ S ✿
n

ai si | ai ∈ A, si ∈ S, n ∈ N

S =

.

i=0

✭✸✮ ▼æ✤✉♥ ♥➔♦ ❝ô♥❣ ❝â ❤➺ s✐♥❤✱ ❝❤➥♥❣ ❤↕♥ M ❝❤➼♥❤ ❧➔ ❤➺ s✐♥❤ ❝õ❛
♠æ✤✉♥ M ✳ ❍➺ s✐♥❤ ❝õ❛ ♠é✐ ♠æ✤✉♥ ❦❤æ♥❣ ❧➔ ❞✉② ♥❤➜t✳
✭✹✮ ❍❛✐ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ❝õ❛ ♠ët ♠æ✤✉♥ ❝â t❤➸ ❦❤æ♥❣ ❝ò♥❣ ❧ü❝
❧÷ñ♥❣✳ ❚❤➟t ✈➟②✱ ①➨t Z ♥❤÷ ❧➔ ♠ët Z ✲ ♠æ✤✉♥✳ ❑❤✐ ✤â {1} ✈➔ {3, 5}
✤➲✉ ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ Z ♥❤÷♥❣ sè ♣❤➛♥ tû ❦❤→❝ ♥❤❛✉✳ ❚✉②
♥❤✐➯♥✱ ✤è✐ ✈î✐ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ t❤➻ s➩ ❦❤→❝✳
❱➼ ❞ö ✷✳✷✳✶✳ ✭✶✮ A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❑❤✐ ✤â A ❧➔ ♠ët
♠æ✤✉♥ tr➯♥ ❝❤➼♥❤ ♥â✳ ❙✉② r❛ A ❧➔ ♠ët ♠æ✤✉♥ ①②❝❧✐❝ s✐♥❤ ❜ð✐ {1} ✈➻
A = {r.1 | r ∈ A} = 1 ✳
✭✷✮ ❈❤♦ U ❧➔ ♠ët K ✲ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝â ❝❤✐➲✉ n ✈î✐ ❝ì sð
S = {e1 , e2 , ..., en }✳ ❑❤✐ ✤â U ❧➔ K ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ S ❧➔ t➟♣
s✐♥❤ ❝õ❛ K ✲ ♠æ✤✉♥ U ✳
✣à♥❤ ❧➼ ✷✳✷✳✶✳ ❈❤♦ ❞➣② ❦❤î♣ ♥❣➢♥ ❝→❝ A ✲ ♠æ✤✉♥✿
f


g

0 → N → M → P → 0.

❚❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿
✐✮ ◆➳✉ M ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ P ❝ô♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳
✐✐✮ ◆➳✉ N ✈➔ P ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ M ❝ô♥❣ ❤ú✉ ❤↕♥ s✐♥❤✳
✷✶


ự sỷ M ỳ s õ s {y1, ..., yn}
g t P ữủ s {g (y1) , ..., g (yn)}
r P ổ ỳ s
N P ỳ s sỷ {x1, ..., xm} {z1, ..., zn}
tữỡ ự s ừ N P g t ợ ồ
zi P tỗ t yi M g (yi ) = zi ợ ồ i = 1, n ự
{f (xi ) , ..., f (xm ) , y1 , ..., yn } ởt s ừ M t tũ
ỵ y M s r g (y) P tỗ t a1, ..., an A s
n

g (y) =

n

ai zi =
i=1

n

ai g (yi ) = g


ai y i

i=1

i=1

g y ni=1 aiyi = 0 s r y
tỗ t x = ni=1 bixi N s

n
i=1 ai yi +

=

i=1
m
i=1 bi f

Kerg = Imf

m

bi xi

ai yi = f (x) = f
i=1

y =
s


n
i=1 ai yi

m

n

y

.

(xi ) M

bi f (xi ) .
i=1

ởt ổ ỳ

ứ tr t õ q s

sỷ N ởt ổ ừ M õ
M ởt A ổ ỳ s t M/N ởt A
ổ ỳ s
N M/N A ổ ỳ s t M ởt
A ổ ỳ s

q

ổ ỳ s tr

A ởt õ ỡ 1 = 0



✭✣à♥❤ ❧➼ ❍❛♠✐❧t♦♥ ✲ ❈❛②❧❡② ♠ð rë♥❣✮✳ ❈❤♦ M ❧➔ ♠ët
♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ A✳ ●✐↔ sû M ❝â ♠ët ❤➺
s✐♥❤ ❣ç♠ n ♣❤➛♥ tû✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ A ✈➔ Φ ❧➔ ♠ët tü ✤ç♥❣ ❝➜✉ ✤ç♥❣
♥❤➜t A ✲ ♠æ✤✉♥ ❝õ❛ M s❛♦ ❝❤♦ Φ (M ) ⊆ IM ✳ ❚❤➳ t❤➻ tç♥ t↕✐ ❝→❝
ai ∈ I i ✈î✐ i = 1, n s❛♦ ❝❤♦ Φn + a1 Φn−1 + ... + an−1 Φ + an = 0✳

✣à♥❤ ❧➼ ✷✳✷✳✸✳

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ❜➜t ❦➻ a ∈ A ✱ →♥❤ ①↕✿
a: M →M
x → ax

❧➔ ♠ët tü ✤ç♥❣ ❝➜✉ ❝õ❛ M ✈➔ ❣å✐ ❧➔ tü ✤ç♥❣ ❝➜✉ ♥❤â♠✳ ❑❤✐ ✤â ✈➔♥❤
❝ì sð A ✤÷ñ❝ ①❡♠ ♥❤÷ ❧➔ ✈➔♥❤ ❝→❝ ✤ç♥❣ ❝➜✉ ♥❤➙♥ ❝õ❛ M ✳ ●✐↔ sû Φ ❧➔
♠ët tü ✤ç♥❣ ❝➜✉ ❜➜t ❦➻ ❝õ❛ M ✱ t❛ ①➨t t➟♣
A [Φ] = bm Φm + bm−1 Φm−1 + ... + b0 | b0 , ...bm ∈ A, m ≥ 0 .

❑❤✐ ✤â A [Φ] ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ❣ç♠ ♥❤ú♥❣ tü
✤ç♥❣ ❝➜✉ ❝õ❛ M ✳ ✣➦t f.x = f (x) ✈î✐ ♠é✐ f ∈ A [Φ] ✈➔ x ∈ M ✳
◆➳✉ M ❧➔ ♠æ✤✉♥ θ t❤➻ Φ ❧➔ ✤ç♥❣ ❝➜✉ θ✱ ✤à♥❤ ❧➼ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
◆➳✉ ♠æ✤✉♥ M ❦❤→❝ θ✱ ❣✐↔ sû M ❝â ❤➺ s✐♥❤ {x1, x2, ..., xn} t❤➻
Φ (xi ) ∈ IM ✳ ❉♦ ✤â tç♥ t↕✐ ❝→❝ aij ∈ I ✈î✐ 1 ≤ i, j ≤ n ✤➸✿


a


−Φ

 11
 a
 21

 ...


an1

a12

...

a1n

a22 − Φ ...

a2n

x

  1
 x 
  2
   = 0.
  ... 
 


...

...

an2

... ann − Φ

✷✸

...

 

xn