Iđêan nguyên tố liên kết (2018)

❚❘×❮◆● ✣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)❀

✶✵

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

✣à♥❤ ♥❣❤➽❛ ✶✳✸✷✳ ❈❤♦ ✤ç♥❣ ❝➜✉ ✈➔♥❤ f : X −→ Y ✳ ❚❛ ♥â✐ f

❧➔ ✤ì♥ ❝➜✉

✭t♦➔♥ ❝➜✉ ❤❛② ✤➥♥❣ ❝➜✉ ✮ ♥➳✉ →♥❤ ①↕ f ❧➔ ✤ì♥ →♥❤ ✭t♦➔♥ →♥❤ ❤❛② s♦♥❣
→♥❤✱ t÷ì♥❣ ù♥❣✮✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ f ❧➔ ✤➥♥❣ ❝➜✉ t❤➻ t❛ ♥â✐ ❤❛✐ ✈➔♥❤ X
✈➔ Y ❧➔ ✤➥♥❣ ❝➜✉ ✈î✐ ♥❤❛✉✱ ❦þ ❤✐➺✉ X ∼
= Y✳

❱➼ ❞ö ✶✳✸✸✳
✭✶✮ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ ✈➔♥❤ R✳ ⑩♥❤ ①↕

A −→ R
a −→ a
❧➔ ♠ët ✤ì♥ ❝➜✉ ✈➔ ❣å✐ ❧➔ ✤ì♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳
✭✷✮ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ⑩♥❤ ①↕

ϕ : R −→ R/I
x −→ x + I
❧➔ ♠ët ✤ç♥❣ ❝➜✉ tø ✈➔♥❤ R ✤➳♥ ✈➔♥❤ t❤÷ì♥❣ R/I ✳ ✣ç♥❣ ❝➜✉ ♥➔②
❝á♥ ❧➔ ♠ët t♦➔♥ ❝➜✉✱ ❣å✐ ❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❍ì♥ ♥ú❛✱ Ker ϕ = I ✳

▼➺♥❤ ✤➲ ✶✳✸✹✳ ❈❤♦ ✤ç♥❣ ❝➜✉ ✈➔♥❤ f : A −→ B ✳ ❑❤✐ ✤â
✶✳ f (0) = 0❀

✷✳ f (−a) = −f (a) ✈î✐ ♠å✐ a ∈ A❀
✸✳ f (a − b) = f (a) − f (b) ✈î✐ ♠å✐ a, b ∈ A✳

▼➺♥❤ ✤➲ ✶✳✸✺✳ ◆➳✉ f : X −→ Y ✱ g : Y −→ Z ❧➔ ❤❛✐ ✤ç♥❣ ❝➜✉ ✈➔♥❤
t❤➻ t➼❝❤ ✭→♥❤ ①↕ ❤ñ♣ t❤➔♥❤✮ g ◦ f : X −→ Z ❝ô♥❣ ❧➔ ♠ët ✤ç♥❣ ❝➜✉ ✈➔♥❤✳
✶✶


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❈→❝ ♠➺♥❤ ✤➲ ❞÷î✐ ✤➙② ❧➔ ♠ët tr♦♥❣ ❝→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛
✤ç♥❣ ❝➜✉ ✈➔♥❤✳

▼➺♥❤ ✤➲ ✶✳✸✻✳ ❈❤♦ f : X −→ Y ❧➔ ♠ët ✤ç♥❣ ❝➜✉ ✈➔♥❤✳ ❑❤✐ ✤â
✭✐✮ ❍↕t ♥❤➙♥ Ker f := f −1 (0) ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R❀
✭✐✐✮ ❷♥❤ Im f = f (X) ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ Y ✳

✣à♥❤ ❧þ ✶✳✸✼✳ ❈❤♦ ✤ç♥❣ ❝➜✉ ✈➔♥❤ f : X −→ Y ✈➔ ρ : X −→ X/ Ker f
❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝ ✳ ❑❤✐ ✤â
✭✐✮ ❚ç♥ t↕✐ ❞✉② ♥❤➜t ✤ç♥❣ ❝➜✉ f¯ : X/ Ker f −→ Y s❛♦ ❝❤♦ ❜✐➸✉ ✤ç s❛✉
❣✐❛♦ ❤♦→♥
f

X

/
f¯

ρ

:

Y

$

X/ Ker f
✭✐✐✮ ✣ç♥❣ ❝➜✉ f¯ ❧➔ ♠ët ✤ì♥ ❝➜✉ ✈➔ Imf¯ = f (X)✳

❍➺ q✉↔ ✶✳✸✽✳ ❱î✐ ♠å✐ ✤ç♥❣ ❝➜✉ ✈➔♥❤ f : X −→ Y t❛ ❝â
f (X) ∼
= X/ Ker f
✣➦❝ tr÷♥❣ ❝õ❛ ✐✤➯❛♥ ♥❣✉②➯♥ tè ✈➔ tè✐ ✤↕✐ ✤÷ñ❝ ❝❤♦ ❜ð✐ ♠➺♥❤ ✤➲ s❛✉
♠➔ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❧➔ sì ❝➜♣✳

▼➺♥❤ ✤➲ ✶✳✸✾✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â
✶✳ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R/I ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥✳
✷✳ I ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R/I ❧➔ ♠ët tr÷í♥❣✳
❍➺ q✉↔ ❧➔ ♠å✐ ✐✤➯❛♥ tè✐ ✤↕✐ ✤➲✉ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳
✶✷


õ tốt ồ

P ồ

tr

R õ ỡ ởt tr

ồ ừ õ ỳ s

ử
số Z tr ồ ừ số

Z õ nZ ỳ s
rữớ số Q, R, C tr {0}
t ừ õ ỳ s

ỵ R õ ỡ õ
s tữỡ ữỡ
R tr
ộ t rộ ừ R ổ tỗ t tỷ ỹ
ợ I1 I2 . . . Ik+q . . . ởt t ừ R
ổ tỗ t n In = In+1 = . . . tự ồ t ừ R
ứ

ổ


R ởt õ ỡ 1 = (M, +) ởt

õ ũ ợ àA ì M M t
ữủ ax = à(ax) ợ ồ a R, x M ợ (M, +, .) ữ
tr t M ữủ ồ Rổ t s tọ ợ ồ

a, b A, x, y M

õ tốt ồ

P ồ

(a + b)x = ax + bx
a(x + y) = ax + ay
a(bx) = (ab)x
1x = x

ử
ộ ừ R ởt Rổ t R ụ ởt

Rổ
ộ ổ tỹ õ ổ ởt Zổ
õ M ữủ Zổ ợ t
ữ s

nx = x + x + ã ã ã + x ợ n Z x M
n>0

nx = (n)(x) ợ n < 0
0x = 0M

ỵ Rổ õ ợ ồ x M a R
0R x = 0M = a0M
(a)x = ax = a(x)

ìợ ừ ổ M

Rổ

P tỷ 0 = a R ồ ữợ ừ ổ tr M tỗ t

0 = x M s ax = 0.
ữủ t tự a ổ ữợ ừ 0 ax = 0 t x = 0
tỷ tr R ữợ ừ 0 tr M ZDR (M ).



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❚➟♣ ❝→❝ ♣❤➛♥ tû tr♦♥❣ R ❦❤æ♥❣ ❧➔ ÷î❝ ❝õ❛ 0 tr♦♥❣ M ❦➼ ❤✐➺✉ ❧➔

N ZDR M ✳ ❚❛ ❝â N ZDR M = R \ ZDR M ✳

✶✳✽ ▼æ✤✉♥ ❝♦♥
✣à♥❤ ♥❣❤➽❛ ✶✳✹✼✳

❈❤♦ M ❧➔ R✕♠æ✤✉♥✳ ❚➟♣ ❝♦♥ N ❝õ❛ M ✤÷ñ❝ ❣å✐ ❧➔

R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ♥➳✉
✭✐✮ N = ∅❀
✭✐✐✮ x − y ∈ N ✈î✐ ♠å✐ x, y ∈ N ❀
✭✐✐✐✮ ax ∈ N ✈î✐ ♠å✐ x ∈ N ✱ a ∈ R✳

❱➼ ❞ö ✶✳✹✽✳

❈❤♦ M ❧➔ R✲♠æ✤✉♥✳

✭✐✮ M ❧✉æ♥ ❝❤ù❛ ✷ ♠æ✤✉♥ ❝♦♥ t➛♠ t❤÷í♥❣ ❧➔ ✭✵✮ ✈➔ M ✳
✭✐✐✮ Rx ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ ▼✱ ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ❝♦♥ ①②❝❧✐❝ s✐♥❤
❜ð✐ x✳
✭✐✐✐✮ ▼å✐ ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ ❆❜❡❧ M ✤➲✉ ❧➔ ♠ët Z✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳
✭✐✈✮ ▼å✐ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R ❝â ✤ì♥ ✈à 1 = 0 ✤➲✉ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ R✳

✣à♥❤ ❧þ ✶✳✹✾✳ ❈❤♦ ▼ ❧➔ R✲♠æ✤✉♥✳ ❚➟♣ N ⊆ M ❧➔ ♠ët R✲♠æ✤✉♥ ❝♦♥
❝õ❛ M ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✭✐✮ 0M ∈ N ❀
✭✐✐✮ ax + by ∈ N ✈î✐ ♠å✐ x, y ∈ N, a, b ∈ R✳

✶✺


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❈❤ù♥❣ ♠✐♥❤✳

⇒] ❍✐➸♥ ♥❤✐➯♥✳
⇐] ❱➻ 0M ∈ N ♥➯♥ N = ∅✳ ❚❛ ❝â✿
x − y = 1x + (−1)y ∈ N ✈î✐ ♠å✐ x, y ∈ N ❀
ax = ax + 0A 0M ∈ N ✈î✐ ♠å✐ a ∈ R, x ∈ M ❀
❉♦ ✤â N ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳

✶✳✾ ▼æ✤✉♥ t❤÷ì♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✺✵✳

❈❤♦ N ❧➔ R✕♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❚➟♣ t❤÷ì♥❣ M/N

❝â ❝➜✉ tró❝ R✕♠æ✤✉♥ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥✿

• ❱î✐ ♠å✐ x + N, y + N ∈ M/N : (x + N ) + (y + N ) = (x + y) + N ❀
• ❱î✐ ♠å✐ x + N ∈ M/N ✱ a ∈ R : a(x + N ) = ax + N ✳
❚❛ ❣å✐ ✤â ❧➔ ♠æ✤✉♥ t❤÷ì♥❣ ❝õ❛ ♠æ✤✉♥ M tr➯♥ ♠æ✤✉♥ ❝♦♥ N ✳

◆❤➟♥ ①➨t ✶✳✺✶✳

◆➳✉ P ❧➔ ♠ët R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✱ P ❝❤ù❛ N t❤➻

P/N ❧➔ ♠ët R✲♠æ✤✉♥ t❤÷ì♥❣ ✈➔ ❝ô♥❣ ❧➔ ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M/N ✳

❱➼ ❞ö ✶✳✺✷✳
✭✐✮ ❱➔♥❤ t❤÷ì♥❣ ❝õ❛ ♠ët ✈➔♥❤ R ❝ô♥❣ ❧➔ ♠ët R✲♠æ✤✉♥ t❤÷ì♥❣ ❝õ❛ R✳
✭✐✐✮ ❚r÷í♥❣ ❝→❝ sè ❤ú✉ t➾ Q ❧➔ ♠ët Z✲♠æ✤✉♥ ✈➔ Z ❧➔ ♠ët Z✲♠æ✤✉♥ ❝♦♥
❝õ❛ Q✱ ❦❤✐ ✤â t❛ ♥❤➟♥ ✤÷ñ❝ Z✲♠æ✤✉♥ t❤÷ì♥❣ Q/Z✱ ❧➔ ♠ët ♠æ✤✉♥
❝❤➾ ❜❛♦ ❣ç♠ ❝→❝ ♣❤➛♥ ❧➫ ❝õ❛ ❝→❝ sè ❤ú✉ t➾✳

✶✻


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

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

❈❤♦ M ✱ M ❧➔ ❤❛✐ R✲♠æ✤✉♥✳ ▼ët →♥❤ ①↕

f : M −→ M ❧➔ ♠ët ✤ç♥❣ ❝➜✉ R✲♠æ✤✉♥ ✭❤❛② →♥❤ ①↕ t✉②➳♥ t➼♥❤✮ ♥➳✉
✭✐✮ f (x + y) = f (x) + f (y) ✈î✐ ♠å✐ x, y ∈ M ❀
✭✐✐✮ f (ax) = af (x) ✈î✐ ♠å✐ a ∈ R ✈➔ ♠å✐ x ∈ M ✳
✣ç♥❣ ❝➜✉ R✕♠æ✤✉♥ ✤÷ñ❝ ❣å✐ ❧➔ ✤ì♥ ❝➜✉✱ t♦➔♥ ❝➜✉ ❤❛② ✤➥♥❣ ❝➜✉ ♥➳✉
→♥❤ ①↕ f t÷ì♥❣ ù♥❣ ❧➔ ✤ì♥ →♥❤✱ t♦➔♥ →♥❤ ❤❛② s♦♥❣ →♥❤✳
⑩♥❤ ①↕ f ✤÷ñ❝ ❣å✐ ❧➔ ✤ç♥❣ ❝➜✉ ❦❤æ♥❣ ♥➳✉ f (M ) = {0M }✳

◆❤➟♥ ①➨t ✶✳✺✹✳

❈❤♦ ♠ët ✤ç♥❣ ❝➜✉ R✲♠æ✤✉♥ f : M −→ M

✭✐✮ f ❧➔ ✤ç♥❣ ❝➜✉ 0 ⇔ Ker f = M ✳
✭✐✐✮ f ❧➔ t♦➔♥ ❝➜✉ ⇔ Im f = M .

❱➼ ❞ö ✶✳✺✺✳
✭✶✮ ❈❤♦ N ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M t❤➻ t❛ ❝â ♠æ✤✉♥ t❤÷ì♥❣ M/N ✳ ❑❤✐
✤â q✉② t➢❝ ρ : M −→ M/N ❝❤♦ ❜ð✐ ρ(x) = x
¯ ❧➔ ♠ët ✤ç♥❣ ❝➜✉

R✲♠æ✤✉♥✳ ❍ì♥ ♥ú❛ ρ ❝á♥ ❧➔ ♠ët t♦➔♥ ❝➜✉✱ ✤÷ñ❝ ❣å✐ ❧➔ t♦➔♥ ❝➜✉
❝❤✐➳✉ ❝❤➼♥❤ t➢❝✳ ❚♦➔♥ ❝➜✉ ♥➔② ❝â Ker ρ = N.
✭✷✮ ❱î✐ ♠é✐ N ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✱ →♥❤ ①↕ ♥❤ó♥❣ ι : N −→ M ❜✐➳♥
♠é✐ ♣❤➙♥ tû ❝õ❛ N t❤➔♥❤ ❝❤➼♥❤ ♥â ❧➔ ♠ët ✤ì♥ ❝➜✉✱ ❣å✐ ❧➔ ✤ì♥ ❝➜✉

❝❤➼♥❤ t➢❝ ❤❛② ♣❤➨♣ ♥❤ó♥❣ ❝❤➼♥❤ t➢❝ tø N ✈➔♦ M ✳

▼➺♥❤ ✤➲ ✶✳✺✻✳ ⑩♥❤ ①↕ f : M −→ M ❧➔ ✤ç♥❣ ❝➜✉ ❝→❝ R✲♠æ✤✉♥ ❦❤✐ ✈➔
❝❤➾ ❦❤✐ f (ax + by) = af (x) + bf (y) ✈î✐ ♠å✐ a, b ∈ R, x, y ∈ M ✳

✶✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠å✐ a, b ∈ R, x, y ∈ M ✳

⇒❪ ●✐↔ sû f ❧➔ ✤ç♥❣ ❝➜✉✳ ❑❤✐ ✤â
f (ax + by) = f (ax) + f (by) = af (x) + bf (y).
⇐] ●✐↔ sû f (ax + by) = af (x) + bf (y). ❚❛ ❦✐➸♠ tr❛ ✷ ✤✐➲✉ ❦✐➺♥✿
f (x + y) = f (1x + 1y) = f (x) + f (y) ❀
f (ax) = f (ax + 0y) = af (x) + 0f (y) = af (x)✳
❱➟② f ❧➔ ✤ç♥❣ ❝➜✉✳

▼➺♥❤ ✤➲ ✶✳✺✼✳ ◆➳✉ f : M −→ N ✱ g : N −→ L ❧➔ ❤❛✐ ✤ç♥❣ ❝➜✉
R✕♠æ✤✉♥ t❤➻ →♥❤ ①↕ t➼❝❤ ✭→♥❤ ①↕ ❤ñ♣✮ g ◦ f : M −→ L ❝ô♥❣ ❧➔ ♠ët ✤ç♥❣
❝➜✉ R✕♠æ✤✉♥✳

✣à♥❤ ❧þ ✶✳✺✽✳ ❈❤♦ f : M −→ M ❧➔ ♠ët ✤ç♥❣ ❝➜✉ ❝→❝ R✕♠æ✤✉♥✳ ❑❤✐
✤â
✭✐✮ ◆➳✉ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M t❤➻ f −1 (N ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥
❝õ❛ M ✳ Ker f ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳
✭✐✐✮ ◆➳✉ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M t❤➻ f (N ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛

M ✳ Im f ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳
✭✐✐✐✮ f ❧➔ ✤ì♥ ❝➜✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Ker f = 0.

✣à♥❤ ❧þ ✶✳✺✾✳ ❈❤♦ f : M −→ N ❧➔ ♠ët ✤ç♥❣ ❝➜✉ ❝→❝ R✲♠æ✤✉♥ ✈➔
p : M −→ M/ Ker f ❧➔ t♦➔♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❑❤✐ ✤â tç♥ t↕✐ ❞✉② ♥❤➜t ✤ì♥

✶✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❝➜✉ f¯ : M/ Ker f −→ N s❛♦ ❝❤♦ ❜✐➸✉ ✤ç s❛✉ ❣✐❛♦ ❤♦→♥
f

M

/
f¯

ρ

9

N

%

M/ Ker f

❍➺ q✉↔ ✶✳✻✵✳ ❈❤♦ f : M −→ N ❧➔ ♠ët ✤ç♥❣ ❝➜✉ ❝→❝ R✲♠æ✤✉♥✳ ❑❤✐ ✤â
t❛ ❝â M/ Ker f ∼
= N✳
= Im f ✳ ◆➳✉ f ❧➔ t♦➔♥ ❝➜✉ t❤➻ M/ Ker f ∼

❍➺ q✉↔ ✶✳✻✶✳

✭✣à♥❤ ❧➼ ✤➥♥❣ ❝➜✉ ◆♦❡t❤❡r t❤ù ♥❤➜t✮✳ ❈❤♦ P

❧➔

♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ N ✱ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M ✳ ❑❤✐
✤â t❛ ❝â ✤➥♥❣ ❝➜✉ M/N ∼
= (M/P )/(N/P )✳
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ✤ç♥❣ ❝➜✉

f : M/P −→ M/N
x + P −→ x + N
❉➵ ❦✐➸♠ tr❛ ✤÷ñ❝ f ❧➔ ♠ët t♦➔♥ ❝➜✉ ✈î✐ Ker f = N/P ✳ ❚❤❡♦ ❍➺ q✉↔ ✶✳✻✵
tr➯♥ t❛ ❝â M/N ∼
= (M/P )/ Ker f ✱ ❞♦ ✤â M/N ∼
= (M/P )/(N/P )✳

❍➺ q✉↔ ✶✳✻✷✳

✭✣à♥❤ ❧➼ ✤➥♥❣ ❝➜✉ ◆♦❡t❤❡r t❤ù ❤❛✐✮

◆➳✉ M, N ❧➔ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❝ò♥❣ ♠ët ♠æ✤✉♥ t❤➻

(M + N )/N ∼
= M/(M ∩ N ).
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ✤ç♥❣ ❝➜✉

f :M −→ (M + N )/N
x −→ f (x) = x¯ = x + N
✶✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

P❤↕♠ ◆❣å❝ ❉✐➺♣

❱î✐ ♠é✐ z¯ = z + N ∈ (M + N )/N ✈î✐ z ∈ M + N ✱ t❛ ❝â z = x + y ✈î✐

x ∈ M, y ∈ N ✳ ❑❤✐ ✤â
z¯ = z + N = (x + y) + N = x + N = x¯
❞♦ ✤â f (x) = z¯✳ ❱➟② f ❧➔ t♦➔♥ ❝➜✉✳
▲↕✐ ❝â Ker f = {x ∈ M | x
¯ = 0} = {x ∈ M | x ∈ N } = M ∩ N ✳ ⑩♣
❞ö♥❣ ❍➺ q✉↔ ✶✳✻✵ t❛ ❝â (M + N )/N ∼
= M/(M ∩ N )✳

✶✳✶✶ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ✈➔♥❤ ✈➔ ♠æ✤✉♥
❈❤♦ M ❧➔ R✲♠æ✤✉♥✳ S ❧➔ t➟♣ ♥❤➙♥ ✤â♥❣ ✭tù❝ ❧➔ S ❝❤ù❛ 1✱ S ❦❤æ♥❣ ❝❤ù❛ 0✱
✈➔ ✈î✐ ♠å✐ a, b ∈ S t❤➻ ab ∈ S ✮✳ ❚r➯♥ t➟♣ M ×S = {(m, s) | m ∈ M, s ∈ S}
t❛ ①→❝ ✤à♥❤ q✉❛♥ ❤➺ ∼ ♥❤÷ s❛✉

(m, s) ∼ (m , s ) ⇔ ∃t ∈ S : t(ms − sm ) = 0
❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ❑❤✐ ✤â t❛ ❦➼ ❤✐➺✉ ❝→❝ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❧➔

(m, s) = {(m , s ) ∈ M × S | (m , s ) ∼ (m, s)} :=

m
s

❚❛ ❦➼ ❤✐➺✉ t➟♣ t❤÷ì♥❣ ❝õ❛ M × S t❤❡♦ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ∼ ❧➔ S −1 M

S −1 M =

m
| m ∈ M, s ∈ S
s

❑❤✐ M = R t❛ ❝â t➟♣

S −1 R =

r
| r ∈ R, s ∈ S
s

✷✵