Iđêan nguyên tố liên kết

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

P❍❸▼ ◆●➴❈ ❉■➏P

■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽


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

P❍❸▼ ◆●➴❈ ❉■➏P

■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿

❚❤❙✳ ✣➱ ❱❿◆ ❑■➊◆

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

▲í✐ ❝↔♠ ì♥
❚r÷î❝ ❦❤✐ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä
❧á♥❣ ❝↔♠ ì♥ tî✐ ❝→❝ t❤➛② ❝æ ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐
✷✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ❜ë ♠æ♥ ✤↕✐ sè ❝ô♥❣ ♥❤÷ ❝→❝ t❤➛② ❝æ t❤❛♠ ❣✐❛
❣✐↔♥❣ ❞↕② ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ tr✐ t❤ù❝ q✉þ ❜→✉ ✈➔ t↕♦ ✤✐➲✉
❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ö ❦❤â❛ ❤å❝ ✈➔ ❦❤â❛ ❧✉➟♥✳
✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tî✐ t❤➛②
❣✐→♦ ✲ ❚❤↕❝ s➽

✣é ❱➠♥ ❑✐➯♥ ✱ ♥❣÷í✐ ✤➣ trü❝ t✐➳♣ ❤÷î♥❣ ❞➝♥✱ ❝❤➾ ❜↔♦

t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❉♦ t❤í✐ ❣✐❛♥✱ ♥➠♥❣ ❧ü❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜↔♥
❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❱➻ ✈➟②✱ ❡♠ r➜t ♠♦♥❣ ♥❤➟♥
✤÷ñ❝ ♥❤ú♥❣ þ ❦✐➳♥ ❣â♣ þ q✉þ ❜→✉ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳

❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❚→❝ ❣✐↔

P❤↕♠ ◆❣å❝ ❉✐➺♣

✐


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

▲í✐ ❝❛♠ ✤♦❛♥
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

✧■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✧ ✤÷ñ❝ ❤♦➔♥

t❤➔♥❤ ❞♦ sü ❝è ❣➢♥❣✱ ♥é ❧ü❝ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈î✐ sü ❣✐ó♣ ✤ï
t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦ ✲ ❚❤↕❝ ❙➽

✣é ❱➠♥ ❑✐➯♥ ✳

❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ♥❤÷ ✤➣
✈✐➳t tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❱➻ ✈➟②✱ ❡♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦➳t q✉↔
tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ trò♥❣ ✈î✐ ❦➳t q✉↔ ❝õ❛ t→❝
❣✐↔ ♥➔♦ ❦❤→❝✳

❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❚→❝ ❣✐↔

P❤↕♠ ◆❣å❝ ❉✐➺♣

✶


▼ö❝ ❧ö❝
▼ð ✤➛✉

✶

✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✹

✶✳✶

❱➔♥❤

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

✶✳✷

❱➔♥❤ ❝♦♥

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

✻

✶✳✸

■✤➯❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✹

❱➔♥❤ t❤÷ì♥❣

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

✶✵

✶✳✺

✣ç♥❣ ❝➜✉ ✈➔♥❤ ✈➔ ❝→❝ ✤à♥❤ ❧➼ ✤ç♥❣ ❝➜✉ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✻

❱➔♥❤ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✼

▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✽

▼æ✤✉♥ ❝♦♥

✶✺

✶✳✾

▼æ✤✉♥ t❤÷ì♥❣

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

✹

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

✶✻

✶✳✶✵ ✣ç♥❣ ❝➜✉ ♠æ✤✉♥ ✈➔ ❝→❝ ✤à♥❤ ❧➼ ✤ç♥❣ ❝➜✉ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✶✶ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ✈➔♥❤ ✈➔ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷ ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚

✷✹

✸ ❙Ü P❍❹◆ ❚➑❈❍ ◆●❯❨➊◆ ❙❒

✸✾

❑➳t ❧✉➟♥

✹✼
✶


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

▼Ð ✣❺❯
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐
❈❤♦ R ❧➔ ♠ët ✈➔♥❤✱ M ❧➔ R✲♠æ✤✉♥✳ ▼ët ✈➜♥ ✤➲ ✤➦t r❛ tr♦♥❣ ✤↕✐
sè ❣✐❛♦ ❤♦→♥ ❧➔ ❦❤✐ ♥➔♦ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè tr♦♥❣ ✈➔♥❤ R ❧➔ ♠ët ✐✤➯❛♥
♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ♠æ✤✉♥ M ✳ ❚ø ✤â ❙❤✐r♦ ●♦t♦ ✲ ♠ët ♥❤➔ ❚♦→♥
❤å❝ ♥❣÷í✐ ◆❤➟t ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤÷❛ r❛ ❝→❝ ❦➳t q✉↔ ✈➲ t➟♣ ❝❤ù❛ t➜t
❝↔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✈î✐ ♠ët ♠æ✤✉♥✳ ◆❣♦➔✐ r❛ æ♥❣ ❝á♥ ♥❣❤✐➯♥
❝ù✉ ✈➲ ✤à♥❤ ❧➼ ❧å❝ ❇♦✉r❜❛❦✐ ✈➔ ❤➺ q✉↔ ❝õ❛ ♥â ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè
❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ P❤➙♥
t➼❝❤ ♥❣✉②➯♥ sì ❝ô♥❣ ❧➔ ♠ët ✤è✐ t÷ñ♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ✤↕✐ sè✳ ❍✐❞❡②✉❦✐
▼❛ts✉♠✉r❛ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ①✉➜t ❜↔♥ ❝✉è♥ s→❝❤ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣
t❤❡♦r②✱ tr♦♥❣ ✤â ✤÷❛ r❛ ❝→❝ ❧➼ t❤✉②➳t ✈➲ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✈➔ ♣❤➙♥
t➼❝❤ ♥❣✉②➯♥ sì✳ ❚➔✐ ❧✐➺✉ ♥➔② ✤÷❛ r❛ ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛
❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r✱ ✤â ❧➔
♠å✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ✤➲✉ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ✈➔ ♠æ✤✉♥ ❝♦♥ ❜➜t
❦❤↔ q✉② t❤➻ ♥❣✉②➯♥ sì✳
◆❤ú♥❣ ✈➜♥ ✤➲ tr➯♥ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✤↕✐ sè ✈➔ ✤÷ñ❝ ♥❤✐➲✉
♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❤➺ t❤è♥❣ ❧↕✐
♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ✤↕✐ sè ❣✐❛♦ ❤♦→♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✈➜♥ ✤➲
♥❣❤✐➯♥ ❝ù✉✱ s❛✉ ✤â tr➻♥❤ ❜➔② ❧↕✐ ❝❤✐ t✐➳t ❝→❝ ✤à♥❤ ❧➼ tr➯♥✳ ❇➯♥ ❝↕♥❤ ✤â
❝ô♥❣ s➩ ✤÷❛ r❛ ❤➺ t❤è♥❣ ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ ❜ê ✤➲✱ ♥❤➟♥ ①➨t ✤➸ ✤÷❛ ✤➳♥ ❝→❝
❦➳t q✉↔ ♥➯✉ tr➯♥✳
✶


õ tốt ồ

P ồ

ố tữủ ự
t ự tr ổ ữỡ
õ ừ ổ tố t t t ừ
õ sỹ t sỡ

Pữỡ ự
ự tr s t t q
ở ự

trú õ
ỗ ữỡ

ữỡ 1 tự
ữỡ 1 tr ởt số tự ỡ s ừ số õ
tữỡ tr ỗ
ổ ổ ổ tữỡ ữỡ õ ừ ổ
ợ tự ử ử ự
ỵ ừ ữỡ s

ữỡ 2 tố t
ữỡ 2 ữ r tố t ũ
q t tố t
r tr ở ữỡ ỏ ợ t ự
ồ r tứ õ ữ r t t ỳ ừ t
tố t ừ ổ tr tr



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

• ❈❤÷ì♥❣ 3✿ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì
◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ 3 tr➻♥❤ ❜➔② ✈➲ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ P❤➛♥
✤➛✉ ❝õ❛ ❣ç♠ ❝â ✤à♥❤ ♥❣❤➽❛ ♠æ✤✉♥ ♥❣✉②➯♥ sì✱ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥
sì ✈➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉②✳ ▼ët ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ✈➲ ♣❤➙♥ t➼❝❤
♥❣✉②➯♥ sì s➩ ✤÷ñ❝ ✤÷❛ r❛ tr♦♥❣ ♣❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣✳

✸


ữỡ

ở ữỡ tr ởt số tự ỡ s ừ số
ử ử ỹ ự t
t ừ ữỡ s P t ừ ữỡ ỗ ởt số
tự P tự ừ ữỡ ỗ
ổ ổ ổ tữỡ ởt số t t ừ ổ
ởt số ữỡ õ ừ ổ ợ s ữủ
ố ừ ữỡ




R ởt t ủ rộ õ R ũ ợ

t ở (+) (.) ữủ ồ ởt
õ tọ s
R ũ ợ ở ởt õ

R ũ ợ ởt ỷ õ
P ố ố ợ ở tự ợ ồ x, y, z R



õ tốt ồ

P ồ

t (x + y)z = xz + yz z(x + y) = zx + zy
R ữủ ồ õ ỡ R ởt õ
R ữủ ồ ởt õ t t

R ữủ ồ õ ỡ R ởt õ


ử Z, Q, R, Z[x]
ú ỵ

r t ở tỷ ổ ừ ổ

ữủ ỵ 0 P tỷ ỡ ừ õ ổ ữủ ỵ
1
s ữ r ởt số t t ỡ tr

R ởt õ
0x = x0 = 0 ợ ồ x R
(x)y = x(y) = (xy) ợ ồ x, y R
(x)(y) = xy ợ ồ x, y R
x(y z) = xy xz; (x y)z = xz yz ợ ồ x, y, z R

(x)2n = x2n ; (x)2n+1 = x2n+1 ợ ồ x R, n N

ìợ ừ ổ R

ởt ồ

tỷ 0 = a R ữợ ừ tỗ t 0 = b R tọ q

ab = 0



ởt õ ỡ tỷ

õ ỡ ổ õ ữợ ừ ữủ ồ



õ tốt ồ
ử

P ồ

số Z ởt

rữớ R õ ỡ õ ỡ
ởt tỷ ữủ ồ trữớ ợ ồ 0 = x R tỗ t
tỷ x1 tọ x1 x = 1

X

ởt A ởt ở ừ X ờ

ợ t tr X x+y A xy A ợ ồ x, y A

A ởt ừ X A ũ ợ t s
tr A ởt

ỵ A ởt ở rộ ừ X
s tữỡ ữỡ
A ởt ừ X
ợ ồ x, y A t x + y A, xy A, x A
ợ ồ x, y A t x y A, xy A

ử
ở {0} ỗ õ tỷ ổ ở X
ừ X
ở mZ ỗ số ở ừ ởt số m
trữợ ởt ừ Z

ừ ởt ồ tũ ỵ rộ ừ ởt
R ởt ừ R



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

✶✳✸ ■✤➯❛♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳ ▼ët ✈➔♥❤ ❝♦♥ I

❝õ❛ ♠ët ✈➔♥❤ R ❧➔ ✐✤➯❛♥ tr→✐ ✭✐✤➯❛♥

♣❤↔✐ ✮ ❝õ❛ ✈➔♥❤ R ♥➳✉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ xa ∈ I (ax ∈ I) ✈î✐ ♠å✐

a ∈ I, x ∈ R✳ ▼ët ✈➔♥❤ ❝♦♥ I ❝õ❛ ✈➔♥❤ R ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ♥➳✉
✈➔ ❝❤➾ ♥➳✉ I ✈ø❛ ❧➔ ✐✤➯❛♥ tr→✐ ✈ø❛ ❧➔ ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ R✳

✣à♥❤ ❧þ ✶✳✶✹✳ ▼ët ❜ë ♣❤➟♥ I ❝õ❛ ♠ët ✈➔♥❤ R ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ♥➳✉
t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉
✭✐✮ I = ∅❀
✭✐✐✮ a − b ∈ I ✈î✐ ♠å✐ a, b ∈ I ❀
✭✐✐✐✮ ax ∈ I ✈➔ xa ∈ I ✈î✐ ♠å✐ x ∈ R, a ∈ I ✳

❱➼ ❞ö ✶✳✶✺✳
✭✐✮ ❇ë ♣❤➟♥ {0} ✈➔ ❜ë ♣❤➟♥ R ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳
✭✐✐✮ ❇ë ♣❤➟♥ mZ ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ Z✳

▼➺♥❤ ✤➲ ✶✳✶✻✳ ●✐❛♦ ❝õ❛ ♠ët ❤å tò② þ ❦❤→❝ ré♥❣ ❝→❝ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤
R ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳

◆❤➟♥ ①➨t ✶✳✶✼✳

◆➳✉ R ❧➔ ♠ët ✈➔♥❤ ❝â ✤ì♥ ✈à ✈➔ ♥➳✉ I ❧➔ ♠ët ✐✤➯❛♥

❝õ❛ R ❝❤ù❛ ✤ì♥ ✈à ❝õ❛ R t❤➻ I = R✳

▼➺♥❤ ✤➲ ✶✳✶✽✳ ❈❤♦ X ✱ Y ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✱ t❛ ✤à♥❤ ♥❣❤➽❛

X + Y := {a + b | a ∈ X, b ∈ Y };
n

ai bi | ai ∈ X, bi ∈ Y, n ∈ N∗

XY :=
i=1

✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❑❤✐ ✤â X + Y, XY ❧➔ ❝→❝ ✐✤➯❛♥ ❝õ❛ R✱ ❣å✐ ❧➔ tê♥❣✱ t➼❝❤ ❝õ❛ ❤❛✐ ✐✤➯❛♥✳

X :R Y = {x ∈ R | xY ⊆ X}
❝ô♥❣ ❧➔ ✐✤➯❛♥ ❝õ❛ R ✈➔ ❣å✐ ❧➔ ✐✤➯❛♥ ❝❤✐❛✱ tr♦♥❣ ✤â xY = {xb | b ∈ Y }✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✾✳

❈❤♦ S ❧➔ ♠ët ❜ë ♣❤➟♥ ❝õ❛ ✈➔♥❤ R✳ ❚❤❡♦ ✣à♥❤

❧➼ ✶✳✶✷✱ ❣✐❛♦ A ❝õ❛ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ❝❤ù❛ S ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛
✈➔♥❤ R ❝❤ù❛ S ✱ ✐✤➯❛♥ ♥➔② ❣å✐ ❧➔ ✐✤➯❛♥ s✐♥❤ r❛ ❜ð✐ S ✱ ❦➼ ❤✐➺✉ ❧➔ S ✳
◆➳✉ S = {a1 , a2 , · · · , an } t❤➻ A ❣å✐ ❧➔ ✐✤➯❛♥ s✐♥❤ r❛ ❜ð✐ ❝→❝ ♣❤➛♥ tû

a1 , a2 , · · · , an ✱ ❦➼ ❤✐➺✉ A = a1 , a2 , · · · , an ✳ ■✤➯❛♥ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû
❣å✐ ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✳

◆❤➟♥ ①➨t ✶✳✷✵✳ X + Y

❧➔ ✐✤➯❛♥ ♥❤ä ♥❤➜t ❝❤ù❛ X ∪ Y ✳ ◆â✐ ❝→❝❤ ❦❤→❝

X +Y = X ∪Y ✳

▼➺♥❤ ✤➲ ✶✳✷✶✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ a1 , a2 , · · · ,
an ∈ R✳ ❑❤✐ ✤â ✐✤➯❛♥ s✐♥❤ ❜ð✐ S = {a1 , a2 , · · · , an } ❝â ❞↕♥❣
n

xi ai | xi ∈ R .

S =
i=1

✣➠❝ ❜✐➺t a = aR ✈î✐ ♠å✐ a ∈ R✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✷✳

▼ët ✐✤➯❛♥ t❤ü❝ sü P ❝õ❛ ✈➔♥❤ R ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥

♥❣✉②➯♥ tè ♥➳✉ xy ∈ P t❤➻ s✉② r❛ x ∈ P ❤♦➦❝ y ∈ P ✳
▼ët ✐✤➯❛♥ t❤ü❝ sü M ❝õ❛ ✈➔♥❤ R ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐ ♥➳✉ ❝❤➾
❝â ❤❛✐ ✐✤➯❛♥ ❝õ❛ R ❝❤ù❛ M ❧➔ M ✈➔ R✳
❚➟♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R ❦➼ ❤✐➺✉ ❧➔ Spec R✳

✽

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❚➟♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R ❝❤ù❛ I ❦➼ ❤✐➺✉ ❧➔ Var(I)✳

Var(I) = {P | P ∈ Spec R, P ⊇ I}

❇ê ✤➲ ✶✳✷✸✳ ✭❇ê

✤➲ ❩♦r♥✮ ❈❤♦ X ❧➔ t➟♣ s➢♣ t❤ù tü ❝â t➼♥❤ ❝❤➜t ✧♠å✐

t➟♣ ❝♦♥ s➢♣ t❤ù tü t♦➔♥ ♣❤➛♥ ✤➲✉ ❝â ❝❤➦♥ tr➯♥ t❤✉ë❝ X ✧✳ ❑❤✐ ✤â X ❝â
♣❤➛♥ tû ❝ü❝ ✤↕✐✳

◆❤➟♥ ①➨t ✶✳✷✹✳ ▼å✐ ✈➔♥❤ ❝â ✤ì♥ ✈à ✤➲✉ ❝â ✐✤➯❛♥ ❝ü❝ ✤↕✐ ✈➔ ❞♦ ✤â ❧✉æ♥
tç♥ t↕✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✺✳

❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â

√

I = {x ∈ R | ∃n ∈ N∗ : xn ∈ I}

❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ I ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ❝➠♥ ❝õ❛ I ✳
❚➟♣

√

I ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R ❝❤ù❛ I tù❝

❧➔✿

√
I=

P
P ∈Spec R,P ⊇I

◆➳✉ P ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè t❤➻

✣à♥❤ ♥❣❤➽❛ ✶✳✷✻✳

√

P = P✳

▼ët ✐✤➯❛♥ t❤ü❝ sü I ❝õ❛ ✈➔♥❤ R ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥

♥❣✉②➯♥ sì ♥➳✉ ✈î✐ ♠å✐ a, b ∈ R, ab ∈ I ✱ a ∈
/ I t❤➻ b ∈

◆❤➟♥ ①➨t ✶✳✷✼✳

√

I✳

◆➳✉ P ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè t❤➻ P ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✽✳

❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â I ✤÷ñ❝ ❣å✐

❧➔ Q✲♥❣✉②➯♥ sì ♥➳✉ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì ✈➔

✣à♥❤ ♥❣❤➽❛ ✶✳✷✾✳

√

I = Q✳

❈❤♦ R ❧➔ ♠ët ✈➔♥❤✳ ✣➦t 0 :R x ❧➔ t➟♣ ❣ç♠ t➜t ❝↔

❝→❝ ♣❤➛♥ tû a ∈ R t❤♦↔ ♠➣♥ ax = 0✳ ❑❤✐ ✤â 0 :R x ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R
✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ❧✐♥❤ ❤♦→♥ tû ❝õ❛ x tr♦♥❣ R✱ ❦➼ ❤✐➺✉ annR (x) ❤♦➦❝ ann(x)✳
❚ù❝ ❧➔

annR (x) = 0 :R x = {a ∈ R | ax = 0} .
✣➦❝ ❜✐➺t annR 1 = 0, annR 0 = R✳
❑❤→✐ ♥✐➺♠ ✐✤➯❛♥ ❝❤♦ t❛ ♠ët ✤è✐ t÷ñ♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ✈➔♥❤ ❧➔

✈➔♥❤ t❤÷ì♥❣✳

✶✳✹ ❱➔♥❤ t❤÷ì♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✸✵✳
❤➺ ❤❛✐ ♥❣æ✐

❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❉➵ t❤➜② r➡♥❣ q✉❛♥

tr➯♥ R ❝❤♦ ❜ð✐ a

b ⇔ a − b ∈ I ✈î✐ ♠å✐ a, b ∈ R ❧➔ ♠ët

q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ✣➦t t➟♣ t❤÷ì♥❣ R/I = {x + I | x ∈ R}✳ ❚r➯♥ R/I
t❛ tr❛♥❣ ❜à ❤❛✐ ♣❤➨♣ t♦→♥

(x + I) + (y + I) = (x + y) + I ✈î✐ ♠å✐ x + I, y + I ∈ R/I
(x + I)(y + I) = xy + I ✈î✐ ♠å✐ x + I, y + I ∈ R/I
❑❤✐ ✤â R/I ❝ò♥❣ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ ♥➔② ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤ ✈➔ ✤÷ñ❝ ❣å✐
❧➔ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ R t❤❡♦ ✐✤➯❛♥ I ✳

✶✳✺ ✣ç♥❣ ❝➜✉ ✈➔♥❤ ✈➔ ❝→❝ ✤à♥❤ ❧➼ ✤ç♥❣ ❝➜✉ ✈➔♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✸✶✳ ❈❤♦ X, Y

❧➔ ❤❛✐ ✈➔♥❤✳ ▼ët →♥❤ ①↕ f : X −→ Y ✤÷ñ❝

❣å✐ ❧➔ ✤ç♥❣ ❝➜✉ ✈➔♥❤ ♥➳✉ ✈î✐ ♠å✐ a, b ∈ X
✭✐✮ f (a + b) = f (a) + f (b)❀
✭✐✐✮ f (ab) = f (a)f (b)❀

✶✵

õ tốt ồ

P ồ

ỗ f : X Y õ f

ỡ

t f ỡ t s
tữỡ ự r trữớ ủ f t t õ X
Y ợ ỵ X
= Y

ử
A ởt ừ R

A R
a a
ởt ỡ ồ ỡ t
I ởt ừ R

: R R/I
x x + I
ởt ỗ tứ R tữỡ R/I ỗ
ỏ ởt t ồ t t ỡ ỳ Ker = I

ỗ f : A B õ
f (0) = 0

f (a) = f (a) ợ ồ a A
f (a b) = f (a) f (b) ợ ồ a, b A

f : X Y g : Y Z ỗ
t t i
n õ tỗ t si
/ Q s
r
1
si f (xi ) = 0 ợ ồ 1 i n
t s = s1 s2 ã ã ã sn t s
/ Q sf (xi ) = 0 ợ ồ 1

i

n.

r sf (x) = 0 ợ ồ x M, õ sf = 0 õ s 0 :R f = Q
t
õ RQ R f = 0 s r RQ HomR (M, N ) = 0
RQ HomR (M, N )
= HomRQ (MQ , NQ ) = 0 t õ MQ = (0)
ứ õ s r Q SuppR M õ AssR HomR (M, N ) SuppR M
AssR HomR (M, N ) SuppR M AssR N



õ tốt ồ

P ồ

ữủ sỷ Q SuppR M AssR N tự Q SuppR M

Q AssR N
Q SuppR M t õ MQ = (0) MQ RQ ổ ỳ s


x1 x2
xn
x1
x2
xn
,
, ã ã ã , õ MQ = RQ + RQ + ã ã ã + RQ
1 1
1
1
1
1

õ ỗ tỹ

h : M MQ


: MQ RQ /QRQ
a1 x 1 a2 x 2
an xn
ai
/ Q)

+
ããã +
+ QRQ , (ai R, si
s1 1
s2 1
sn 1
si
t
t

g = h : M MQ RQ /QRQ
ủ t
õ Im g RQ /QRQ =

bi + QRQ | bi RQ , i = 1, n tr õ bi =

ci
, ci R, ti
/ Q t t = t1 t2 ã ã ã tn
/ Q õ t Im g R/Q
ti
g tg t õ t sỷ Im g R/Q õ g t ởt


g : M R/Q
x g(x)

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❱➻ Q ∈ AssR N ♥➯♥ tç♥ t↕✐ ✤ì♥ ❝➜✉ ψ : R/Q −→ N ✳ ✣➦t
g

ψ

λ = ψ ◦ g : M −→ R/Q −→ N
x −→ g(x)
❑❤✐ ✤â 0 :R λ = Q✱ s✉② r❛ Q ∈ AssR Hom(M, N )✳

❇ê ✤➲ ✷✳✶✽✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ Q
❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â ♥➳✉ AssR M = {Q} t❤➻ ✐✤➯❛♥ I = (0) :R M
❧➔ Q✲♥❣✉②➯♥ sì✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤

√

I = Q ✈➔ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì✳

❚❤❡♦ ❇ê ✤➲ ✷✳✶✹ t❛ ❝â SuppR M = Var(annR M ) = Var(I)✳ ❑❤✐ ✤â
✈î✐ ♠å✐ P ∈ SuppR M t❤➻ P ∈ Var(I) ♥➯♥ P ⊇ I ✳
❱➻ P ∈ SuppR M ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✷ t❤➻ tç♥ t↕✐ Q ∈ AssR M s❛♦
❝❤♦ P ⊇ Q✳ ❉♦ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛

√

I t❤➻

√

I ⊇ Q✳

▼➦t ❦❤→❝ Q = AssR M ♥➯♥ tç♥ t↕✐ 0 = x ∈ M ✤➸ Q = annR x ⊇

√
I ✳ ❱➟② Q = I ✳
√
❱î✐ a, b ∈ R, ab ∈
/ I, b ∈ I t❛ ❝❤ù♥❣ ♠✐♥❤ a ∈ I ✳

annR M = I ❞♦ ✈➟② Q ⊇ I ⇒ Q ⊇

√

❱➻ ab ∈ I ♥➯♥ abM = 0✳ ▼➔ bM = 0 ✭❞♦ b ∈
/ I ✮ s✉② r❛ a ❧➔ ÷î❝
❝õ❛ 0 tr➯♥ bM ♥➯♥ ❝ô♥❣ ❧➔ ÷î❝ ❝õ❛ 0 tr➯♥ M ✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻ t❤➻

P ♠➔ AssR M = {Q} ♥➯♥ a ∈ Q =

ZDR (M ) =

√

I ✳ ❉♦ ✤â I ❧➔

P ∈AssR M

✐✤➯❛♥ ♥❣✉②➯♥ sì✳
❱➟② I ❧➔ Q✲♥❣✉②➯♥ sì✳

✣à♥❤ ❧þ ✷✳✶✾✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ❦❤→❝ (0)✳ ❑❤✐ ✤â
❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
✭✶✮ ❚➟♣ AssR M ❝â ❞✉② ♥❤➜t ✶ ♣❤➛♥ tû✳
✸✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

✭✷✮ ❱î✐ ♠å✐ a ∈ R✱ →♥❤ ①↕ ♥❤➙♥

a
ˆ : M −→ M
x −→ ax
❧➔ ✤ì♥ ❝➜✉ ❤♦➦❝ ❧ô② ❧✐♥❤ ✤à❛ ♣❤÷ì♥❣ ✭tù❝ ❧➔ ✈î✐ ♠é✐ x ∈ M tç♥ t↕✐

n ∈ N∗ s❛♦ ❝❤♦ an x = 0✮✳
❈❤ù♥❣ ♠✐♥❤✳

(1) ⇒ (2)✳ ●✐↔ sû AssR M ❝â ❞✉② ♥❤➜t ✶ ♣❤➛♥ tû✳ ✣➦t AssR M = {Q}✳
❱î✐ ❜➜t ❦➻ a ∈ R✱ ♥➳✉ a ∈
/ Q t❤➻ t❤❡♦ ❍➺ q✉↔ ✷✳✼ t❤➻ a ❦❤æ♥❣ ❧➔ ÷î❝ ❝õ❛

0 tr➯♥ M ✳ ❉♦ ✤â a
ˆ ❧➔ ✤ì♥ ❝➜✉✳

◆➳✉ a ∈ Q✱ t❛ ❝❤ù♥❣ ♠✐♥❤ a
ˆ ❧➔ ❧ô② ❧✐♥❤ ✤à❛ ♣❤÷ì♥❣✳ ❚❤➟t ✈➟②✱ ✈î✐
♠å✐ 0 = x ∈ M ✤➦t N = Rx ⊆ M ✱ I = (0) :R N = {a ∈ R | aN = 0}✳
❚❛ ❝â ∅ = AssR N ⊆ AssR M = {Q}✳ ❉♦ ✤â AssR N = {Q}✳
❍ì♥ ♥ú❛ N ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ♥➯♥ t❛ ❝â I = (0) :R N ✈➔ ❧➔

√
Q✲♥❣✉②➯♥ sì✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✽ t❤➻ I = Q✳
√
❉♦ ✤â a ∈ I ♥➯♥ tç♥ t↕✐ n > 0 s❛♦ ❝❤♦ an ∈ I ♥➯♥ an x = 0✳

(2) ⇒ (1)✳ ❱➻ M = 0 ♥➯♥ AssR M = ∅✳ ❱î✐ P, Q ∈ AssR M t❛ ❝➛♥
❝❤ù♥❣ ♠✐♥❤ P = Q✳ ❚❤➟t ✈➟②✳
❱➻ Q ∈ AssR M ♥➯♥ tç♥ t↕✐ 0 = x ∈ M s❛♦ ❝❤♦ Q = (0) :R x✳
❱î✐ ♠å✐ a ∈ P t❛ ❝❤ù♥❣ ♠✐♥❤ a ∈ Q✳ ❱➻ P ∈ AssR M ♥➯♥ a ❧➔ ÷î❝ ❝õ❛

0 tr➯♥ M ❞♦ ✤â a
ˆ ❦❤æ♥❣ t❤➸ ❧➔ ✤ì♥ ❝➜✉✳
❉♦ ✈➟② →♥❤ ①↕ ♥❤➙♥ a
ˆ ❧➔ ❧ô② ❧✐♥❤ ♥➯♥ tç♥ t↕✐ n > 0 ✤➸ an x = 0✳ ❙✉② r❛

an ⊆ 0 :R x = Q✳ ▼➔ P ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➯♥ s✉② r❛ a ∈ Q✳
❉♦ ✤â P ⊆ Q✳ ❱➻ P, Q ✈❛✐ trá ♥❤÷ ♥❤❛✉ ♥➯♥ t÷ì♥❣ tü t❛ ❝â Q ⊆ P ✳
❱➟② P = Q✳
✸✽


ữỡ
ĩ P
ở ố ừ ữỡ ợ t sỹ sỡ tr

P ừ ữỡ ổ
sỡ tữỡ ữỡ õ P t t õ ở
ỗ q t q ừ ởt ổ
tứ õ t t sỡ t t
q ởt q trồ ố q t q sỡ
ữủ r ố ừ ữỡ



R ởt M Rổ N Rổ

ừ M õ N ổ sỡ ừ M ợ ồ

x M \ N, a R tọ ax N t tỗ t n N s
an M N

M Rổ N ổ sỡ ừ M
õ s tữỡ ữỡ
ợ ồ x M \ N, a R tọ ax N t tỗ t n N s
an M N
a R ữợ ừ 0 tr M/N t a


annR (M/N )


õ tốt ồ

P ồ

ự

(1) (2) sỷ õ (1) a ữợ ừ 0 tr M/N
õ tỗ t 0 = x + N M/N s a(x + N ) = N t t õ

x
/ N ax N (1) t s r n N s an M N s
r an M/N = 0 tr M/N õ a

annR (M/N )

(2) (1) sỷ õ (2) õ ợ ồ x M \ N, a N tọ
ax N t õ x + N = 0 tr M/N x
/ N
a(x + N ) = ax + N = 0 tr M/N s r a ữợ ừ 0
tr M/N (2) t a

annR (M/N ) tự n N s

an M/N = 0 õ an M N

ỵ R tr M Rổ ỳ s
õ N M ổ sỡ t AssR M/N
õ ởt tỷ tự |AssR M/N | = 1 ỡ ỳ AssR M/N = {P }

annR (M/N ) = I t I P sỡ I = P
ự sỷ AssR M/N = {P }
õ SuppR M/N = Var(annR (M/N )) = Var(P )
õ P =

annR (M/N ) Var(I) = Var(J)



I=



J

sỷ a ữợ ừ 0 tr M/N õ

a ZDR (M/N ) =

Q = {P } a P a

annR (M/N )

QAssR M

N ổ sỡ ừ M
ữủ sỷ N ổ sỡ ừ M t N = M
õ AssR M/N = t I = annR (M/N ) s r I P sỡ
õ



I = P ợ P Spec R AssR M/N = {P }

t ợ Q AssR M/N t t õ ợ a Q t a ZDR (M/N )

s r a

annR (M/N ) =



I õ Q





I


õ tốt ồ

P ồ

t Q AssR M/N tỗ t 0 = x
M/N tọ Q =

annR x annR (M/N ) = I õ Q

Q = I õ |AssR M/N | = 1



I

ố ũ t r I sỡ t ợ ồ

a, b R, ab I, b
/ I t t õ ab(M/N ) = 0 b(M/N ) = 0 õ

a ZDR (M/N ) s r a annR (M/N ) = I



M Rổ N Rổ sỡ

ừ M AssR M/N = {P } õ P =

annR (M/N ) t t õ r

N ổ P sỡ ừ M

ỵ N N ổ P sỡ ừ M t
N N ụ ổ P sỡ
ự ứ ỡ

M/(N N ) M/N M/N
x + N N (x + N, x + N )
r

= AssR M/(N N ) AssR M/N AssR M/N = {P }.
r

AssR M/(N N ) = {P }.
s r ự

N Rổ ừ M õ N q

N = N1 N2 tr õ N1 , N2 ổ tỹ sỹ ừ N r



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❧↕✐ t❤➻ N ❜➜t ❦❤↔ q✉②✳

◆❤➟♥ ①➨t ✸✳✼✳ N

❜➜t ❦❤↔ q✉② ♥➳✉ N = N1 ∩ N2 t❤➻ N = N1 ❤♦➦❝

N = N2 ✳

◆❤➟♥ ①➨t ✸✳✽✳

◆➳✉ ♠æ✤✉♥ (0) ❧➔ ❦❤↔ q✉② tr♦♥❣ M/N t❤➻ N ❦❤↔ q✉②

tr♦♥❣ M ✳
❚❤➟t ✈➟②✳ ❱➻ ✭✵✮ ❧➔ ❦❤↔ q✉② tr♦♥❣ M/N ♥➯♥ t❛ ❝â ❜✐➸✉ ❞✐➵♥

(0) = M1 /N ∩ M2 /N
tr♦♥❣ ✤â M1 , M2 = N ✳ ❉♦ ✤â N = M1 ∩ M2 ♥➯♥ N ❧➔ ❦❤↔ q✉② tr♦♥❣ M ✳

▼➺♥❤ ✤➲ ✸✳✾✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳
❑❤✐ ✤â ♠å✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✤➲✉ ✈✐➳t ✤÷ñ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ♠æ✤✉♥
❝♦♥ ❜➜t ❦❤↔ q✉②✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t ❚✲ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ♠➔ ❦❤æ♥❣ ❝â
❜✐➸✉ ❞✐➵♥ ♥❤÷ tr♦♥❣ ▼➺♥❤ ✤➲ ✸✳✾✳
◆➳✉ ❚
✲ = ∅ t❤➻ ♥â ❝â ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐ N0 t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠
✭✈➻ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ t❤❡♦ ❇ê ✤➲ ❩♦r♥✮✳ ❱➻ N ∈ ❚
✲ ♥➯♥ N0 ❦❤↔ q✉②
tù❝ ❧➔ N = N1 ∩ N2 ✈î✐ N1 , N2 = N ❞♦ ✤â N

N1 , N2 ✳ ❉♦ t➼♥❤ ❝ü❝ ✤↕✐

❝õ❛ N0 s✉② r❛ N1 , N2 ∈
/ ❚✲ ♥❣❤➽❛ ❧➔ N1 , N2 ✤➲✉ ✈✐➳t ✤÷ñ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛
❤ú✉ ❤↕♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ♥➯♥ N0 ❝ô♥❣ ✈➙② ✭♠➙✉ t❤✉➝♥✮✳ ❱➟②
❚
✲ = ∅✳

❈❤ó þ ✸✳✶✵✳

❙ü ❜✐➸✉ ❞✐➵♥ ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❧➔

❦❤æ♥❣ ❞✉② ♥❤➜t✳

❱➼ ❞ö ✸✳✶✶✳

❈❤♦ R ❧➔ ♠ët tr÷í♥❣ ✈➔ M ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈➨❝tì n ❝❤✐➲✉

tr➯♥ R✳ ❑❤✐ ✤â ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❝õ❛ M ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥
✹✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

✈➨❝tì ❝♦♥ n − 1 ❝❤✐➲✉ ✳ ❘ã r➔♥❣ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈➨❝tì tì n − 2 ❝❤✐➲✉ ❝â
t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ n − 1 ❝❤✐➲✉ t❤❡♦ ♥❤✐➲✉
❝→❝❤ ❦❤→❝ ♥❤❛✉✳

✣à♥❤ ♥❣❤➽❛ ✸✳✶✷✳

❚➟♣ N ❝â ❜✐➸✉ ❞✐➵♥

N = N1 ∩ N2 · · · ∩ Nr (∗)
tr♦♥❣ ✤â N =

Nj ✈î✐ ♠å✐ i = 1, r✳ ◆➳✉ Ni ❧➔ ❜➜t ❦❤↔ q✉② ✭♥❣✉②➯♥
j=i,i=1,r

sì✮ t❤➻ (∗) ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉② t❤✉ ❣å♥ ✭♣❤➙♥ t➼❝❤ ♥❣✉②➯♥

sì t❤✉ ❣å♥✱ t÷ì♥❣ ù♥❣✮ ❝õ❛ N ✳

◆❤➟♥ ①➨t ✸✳✶✸✳

❈❤♦ N = N1 ∩ N2 · · · ∩ Nr ❧➔ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥

sì t❤✉ ❣å♥ ❝õ❛ N ✈î✐ AssR M/Ni = {Pi } ✈î✐ i = 1, r ✳ ◆➳✉ Pi = Pj t❤➻
t❤❡♦ ✤à♥❤ ❧➼ ✸✳✺ Ni ∩ Nj ❝ô♥❣ ❧➔ ♠æ✤✉♥ ❝♦♥ Pi ✲♥❣✉②➯♥ sì✳ ❇ð✐ ✈➟② ♥❤â♠
❝→❝ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì t÷ì♥❣ ù♥❣ ❝ò♥❣ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè t❛ ♥❤➟♥
✤÷ñ❝ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ❝õ❛ N ♠➔ Pi = Pj ✈î✐ ♠å✐ i = j ✱ ❣å✐ ❧➔ ♣❤➙♥
t➼❝❤ ♥❣✉②➯♥ sì ♥❣➢♥ ♥❤➜t ✭❝ô♥❣ ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì t❤✉ ❣å♥ ❝õ❛ N ✮✳

❇ê ✤➲ ✸✳✶✹✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ N
❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ◆➳✉ N ♥❣✉②➯♥ sì tr♦♥❣ M t❤➻ ♠æ✤✉♥ ❝♦♥ (0) ❧➔
♥❣✉②➯♥ sì tr♦♥❣ M/N ✳
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ N ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì tr♦♥❣ M ♥➯♥ ✈î✐ ♠å✐ ♣❤➛♥
tû 0 = a ∈ R ❧➔ ÷î❝ ❝õ❛ 0 tr♦♥❣ M/N t❤➻ t❛ ❝â a ∈

annR (M/N )✳

▼➔ M/N = (M/N )/(N/N ) ♥➯♥ ✈î✐ ♠å✐ a ∈ R ❧➔ ÷î❝ ❝õ❛ 0 tr♦♥❣

(M/N )/(N/N ) t❤➻ a ∈

annR (M/N )/(N/N )✳ ❑❤✐ ✤â N/N ❧➔ ♠æ✤✉♥

❝♦♥ ♥❣✉②➯♥ sì ❝õ❛ M/N ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ ♠æ✤✉♥ (0) ❧➔ ♠æ✤✉♥ ❝♦♥
♥❣✉②➯♥ sì ❝õ❛ M/N ✳
✹✸


õ tốt ồ

P ồ

ỵ R tr M Rổ ỳ s

õ
ởt ổ t q ừ M ổ sỡ
N = N1 N2 ã ã ã Nr t sỡ t ồ ừ
ổ tỹ sỹ N ừ M AssR M/Ni = {Pi } t AssR M/N =

{P1 , P2 , ã ã ã , Pr }
ồ ổ tỹ sỹ N ừ M õ t sỡ
ự sỷ N ổ ừ M

(1). ự N ổ sỡ tr M t N q
tr M õ t

N ổ sỡ tr M ổ (0) ổ sỡ tr
M/N s r tứ ờ
ổ (0) q tr M/N N q tr M s r
tứ t
õ t õ t sỷ N = (0) tự sỷ (0) ổ ổ
sỡ tr M õ |AssR M | = 1 M = (0) t s r

|AssR M |

2 sỷ P1 , P2 AssR M, P1 = P2 tỗ t ỡ
f

g

R/P1 M, R/P2 M
õ tỗ t K1 , K2 M s K1
= R/P1 , K2
= R/P2

õ K1 , K2 = 0 ỡ ỳ K1 K2 = (0) t tỗ t 0 = a + P1

R/P1 , 0 = b + P2 R/P2 tọ 0 = f (a + P1 ) = g(b + P2 ) K1 K2 .