Iđêan trong vành đa thức

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

◆●❯❨➍◆ ❚❍➚ ❚❍Ò❨ ▲■◆❍

■✣➊❆◆ ❚❘❖◆● ❱⑨◆❍ ✣❆ ❚❍Ù❈

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽


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

◆●❯❨➍◆ ❚❍➚ ❚❍Ò❨ ▲■◆❍

■✣➊❆◆ ❚❘❖◆● ❱⑨◆❍ ✣❆ ❚❍Ù❈
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝
❚❤❙✿ ✣➱ ❱❿◆ ❑■➊◆

❍➔ ◆ë✐ ✕ ✷✵✶✽

õ tốt ồ

ũ

ớ ỡ

ởt tớ ự tú t ũ ợ sỹ
ú ù t t ừ t ổ õ tốt
ừ ữủ t tọ ỏ ỡ s
s t ổ tr trữớ ồ ữ P ở
ở ú ù tr q tr õ
t t ỡ t s ộ
ữớ trỹ t ữợ t õ
ỏ tự tớ ừ t ỳ
tr tr õ ổ tr ọ ỳ t sõt
rt ữủ ỳ ỵ õ õ tứ t ổ

ởt ỳ t ỡ
ở t


ũ




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

▲í✐ ❝❛♠ ✤♦❛♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ■✤➯❛♥ tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤
❧➔ ❦➳t q✉↔ ❝õ❛ ❜↔♥ t❤➙♥ ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
❇➯♥ ❝↕♥❤ ✤â✱ ❡♠ ♥❤➟♥ ✤÷ñ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❝æ
❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦
✲❚❤✳s ✣é ❱➠♥ ❑✐➯♥✳
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤æ♥❣ trò♥❣ ✈î✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝✳
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✷


▼ö❝ ❧ö❝
✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð

✶✳✶ ❱➔♥❤✱ ✈➔♥❤ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ▼✐➲♥ ♥❣✉②➯♥ ✈➔ tr÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸ ■✤➯❛♥ ✈➔ ✤ç♥❣ ❝➜✉ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✶ ■✤➯❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ✣ç♥❣ ❝➜✉ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹ ❱➔♥❤ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹✳✶ ❳➙② ❞ü♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥
✶✳✹✳✷ ❚➼♥❤ ❝❤➜t ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ ✳ ✳ ✳
✶✳✺ ❱➔♥❤ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ✈➔♥❤ ✈➔ ♠æ✤✉♥ ✳ ✳ ✳

✳
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✼

✼
✾
✶✶
✶✶
✶✸
✶✹
✶✹
✶✻
✶✼
✶✽

✷ ❇❆❖ ✣➶◆● ◆●❯❨➊◆ ❈Õ❆ ■✣➊❆◆

✷✷

✸ ■✣➊❆◆ ✣❒◆ ❚❍Ù❈

✸✸

✷✳✶ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✷ ❇❛♦ ✤â♥❣ ♥❣✉②➯♥ q✉❛ r❡❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✶ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✶✳✶ K−❝ì sð ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✶✳✷ ❚➟♣ s✐♥❤ ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✸✳✷ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷✳✶ ❈→❝ ♣❤➨♣ t♦→♥ ✤↕✐ sè ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷✳✷ ❇➣♦ ❤á❛ ✈➔ ❝➠♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✸ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ✈➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t
✸✳✸✳✶ ■✤➯❛♥ ✤ì♥ t❤ù❝ ❜➜t ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✸✳✷ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✳
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✳
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✳

✸✼
✸✼
✸✾
✹✶

✹✶
✹✺


▲í✐ ♥â✐ ✤➛✉
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐

❈❤ó♥❣ t❛ ❜✐➳t r➡♥❣ ❤➛✉ ❤➳t ♠å✐ ♥❣➔♥❤ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐ ♥❣➔② ♥❛②
tr♦♥❣ q✉→ tr➻♥❤ ♣❤→t tr✐➸♥ ✤➲✉ ❝➛♥ tî✐ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè✳ ❱➻ ✤↕✐ sè
❜✐➸✉ ❤✐➺♥ rã ♥❤➜t ❤❛✐ ✤➦❝ tr÷♥❣ ❝ì ❜↔♥ ❝õ❛ t♦→♥ ❤å❝ ✤â ❧➔ t➼♥❤ trø✉
t÷ñ♥❣ ✈➔ t➼♥❤ tê♥❣ q✉→t✳ ❇➯♥ ❝↕♥❤ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ♥❤÷ ♥❤â♠✱
✈→♥❤✱ tr÷í♥❣✳✳✳ ✐✤➯❛♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè✳
❱➻ ✈➟② ✈î✐ ♥✐➲♠ s❛② ♠➯ ❚♦→♥ ❤å❝ ✈➔ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥
✈➲ ❜ë ♠æ♥ ♥➔②✱ ❞÷î✐ ❣â❝ ✤ë ♠ët s✐♥❤ ✈✐➯♥ s÷ ♣❤↕♠ t♦→♥ ✈➔ tr♦♥❣ ♣❤↕♠
✈✐ ❝õ❛ ♠ët ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝ò♥❣ ✈î✐ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛
❚❤s✳ ✣é ❱➠♥ ❑✐➯♥ ❡♠ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t ❝õ❛ ♠➻♥❤ ✈➲ ✤➲
t➔✐ ✧■✤➯❛♥ tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ✧✳
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉

◆❣❤✐➯♥ ❝ù✉ ✈➲ ✐✤➯❛♥ tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝✱ ❝ö t❤➸ ❧➔ t➻♠ ❤✐➸✉ ❝→❝
✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥✱ ✤à♥❤ ❧➼ ❝õ❛ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥ ❝õ❛ ✐✤➯❛♥
✈➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝✳
✸✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

▲þ t❤✉②➳t ✈➲ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥ ✈➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝✳

✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

✰✮ P❤➙♥ t➼❝❤ t➔✐ ❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥
✺

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✰✮ ❚ê♥❣ ❤ñ♣ ❦✐♥❤ ♥❣❤✐➺♠ ❝õ❛ ❜↔♥ t❤➙♥✳
✺✳ ❈➜✉ tró❝ ❝õ❛ ❦❤â❛ ❧✉➟♥

❑❤â❛ ❧✉➟♥ ✤÷ñ❝ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝ì sð
❈❤÷ì♥❣ ✷✳ ❇❛♦ ✤â♥❣ ♥❣✉②➯♥ ❝õ❛ ✐✤➯❛♥
❈❤÷ì♥❣ ✸✳ ■✤➯❛♥ ✤ì♥ t❤ù❝✳

✻


❈❤÷ì♥❣ ✶
❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❡♠ ✤÷❛ r❛ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè ✤➸
♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t
❝õ❛ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ✣â ❧➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✱ t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧➼ ❝õ❛
✈➔♥❤✱ ✈➔♥❤ ✤❛ t❤ù❝✱ ✐✤➯❛♥✱ ♠æ✤✉♥✳✳✳

✶✳✶ ❱➔♥❤✱ ✈➔♥❤ ❝♦♥
❈❤♦ t➟♣ X = ∅✱ X ❝ò♥❣ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ ❤❛✐
♥❣æ✐ ❣å✐ ❧➔ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥✱ ❦➼ ❤✐➺✉ t❤❡♦ t❤ù tü ❧➔ ” + ”, ”.”
✤÷ñ❝ ❣å✐ ❧➔ ✈➔♥❤ ♥➳✉ ♥â t❤ä❛ ♠➣♥✿

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳

• X

❝ò♥❣ ✈î✐ ♣❤➨♣ ❝ë♥❣ ❧➔ ♥❤â♠ ❛❜❡♥✳

• X

❝ò♥❣ ✈î✐ ♣❤➨♣ ♥❤➙♥ ❧➔ ♥û❛ ♥❤â♠✳

•

P❤➨♣ ♥❤➙♥ ♣❤➙♥ ♣❤è✐ ✤è✐ ✈î✐ ♣❤➨♣ ❝ë♥❣✱ tù❝ ❧➔ ∀x, y, z ∈ X :
x(y + z) = xy + xz, (x + y)z = xz + yz.

❈❤ó þ ✶✳✶✳✷✳

✲ P❤➛♥ tû tr✉♥❣ ❧➟♣ ❝õ❛ ♣❤➨♣ ❝ë♥❣ ❦➼ ❤✐➺✉ ❧➔ 0✱ ❣å✐ ❧➔
✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

♣❤➛♥ tû ❦❤æ♥❣ ❝õ❛ ✈➔♥❤✳
✲ P❤➛♥ tû ✤ì♥ ✈à ❝õ❛ ♣❤➨♣ ♥❤➙♥ ✭♥➳✉ ❝â✮ ❦➼ ❤✐➺✉ ❧➔ ✶ ❤♦➦❝ ❡✳
✲ ◆➳✉ ♣❤➨♣ ♥❤➙♥ ❝â ✤ì♥ ✈à t❤➻ X ❣å✐ ❧➔ ✈➔♥❤ ❝â ✤ì♥ ✈à✳
✲ ◆➳✉ ♣❤➨♣ ♥❤➙♥ ❣✐❛♦ ❤♦→♥ t❤➻ X ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✳
❱➼ ❞ö ✶✳✶✳✸✳ • ❈→❝ t➟♣ Z✱ Q✱ R✱ C ❧➔ ❝→❝ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥
✈à ✈î✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♥❤➙♥ ❝→❝ sè t❤æ♥❣ t❤÷í♥❣✳

❚➟♣ ❤ñ♣ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n (n ∈ N∗) ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥
❝ë♥❣ ✈➔ ♥❤➙♥ ♠❛ tr➟♥ ❧➔ ✈➔♥❤ ❝â ✤ì♥ ✈à✱ ❦❤æ♥❣ ❣✐❛♦ ❤♦→♥✳
❚➼♥❤ ❝❤➜t ✶✳✶✳✹✳ ❈❤♦ X ❧➔ ♠ët ✈➔♥❤✳ ❑❤✐ ✤â✿
• x0 = 0 = 0x✱ ∀x ∈ X ✳
•

•

◆➳✉ ✈➔♥❤ X ❝â ➼t ♥❤➜t ❤❛✐ ♣❤➛♥ tû t❤➻ 0 = 1✳

• (nx)y = nxy = x(ny) ✱ ∀x, y ∈ X ✱ ∀n ∈ Z✳
• (x − y)z = xz − yz ✱ ∀x, y, z ∈ X ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❈❤♦ X

❧➔ ♠ët ✈➔♥❤✱ t➟♣ ❝♦♥ S ❝õ❛ X ✤÷ñ❝ ❣å✐ ❧➔
t➟♣ ❝♦♥ ♥❤➙♥ ✤â♥❣ ♥➳✉ t❤ä❛ ♠➣♥✿
✐✮ 1 ∈ S ✳
✐✐✮ ❱î✐ ♠å✐ x, y ∈ S t❤➻ xy ∈ S ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ❈❤♦ X ❧➔ ♠ët ✈➔♥❤✱ A ❧➔ ♠ët ❜ë ♣❤➟♥ ❝õ❛ X ê♥
✤à♥❤ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr♦♥❣ X ✱ tù❝ ❧➔
∀x, y ∈ A : x + y ∈ A, x.y ∈ A.

❑❤✐ ✤â✱ ♥➳✉ A ❝ò♥❣ ❤❛✐ ♣❤➨♣ t♦→♥ ❝↔♠ s✐♥❤ tr➯♥ A ❧➔ ♠ët ✈➔♥❤ t❤➻ A
❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ X ✳
✽


õ tốt ồ
ử

{0}




X

ũ

ởt X ổ õ t tữớ

õ ỡ A ừ tự A[x]
A[x] ừ tự A[x, y]

X ởt = A X õ A ởt
ừ X



x, y A : x y A, x.y A.

ỵ s ữủ s r tứ
ừ ởt ồ t ý rộ ỳ
ừ ởt X ởt ừ X
ỵ

trữớ

X

X ởt a, b tỷ tở

õ
a ữủ ồ ở ừ b b ữợ ừ ỵ b|a
c X : a = bc
a = 0 ữủ ồ ữợ ừ 0 d X d = 0 : ad = 0
a ữủ ồ c X : ac = 1
õ a, b t ợ tỗ t tỷ u :
a = ub b = ua
a ữợ ừ b a ổ a ổ t ợ
b t a ữủ ồ ữợ tỹ sỹ ừ b



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✈✐✮ ◆➳✉ a = 0✱ a ❦❤æ♥❣ ❦❤↔ ♥❣❤à❝❤✱ a ❦❤æ♥❣ ❝â ÷î❝ t❤ü❝ sü t❤➻ a
❣å✐ ❧➔ ♣❤➛♥ tû ❜➜t ❦❤↔ q✉②✳
❱➔♥❤ ❣✐❛♦ ❤♦→♥ X ❣å✐ ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ♥➳✉ X ❝â
✤ì♥ ✈à✱ ❝â ♥❤✐➲✉ ❤ì♥ ✶ ♣❤➛♥ tû ✈➔ ❦❤æ♥❣ ❝â ÷î❝ ❝õ❛ 0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳

❱➼ ❞ö ✶✳✷✳✸✳

❈→❝ ✈♥❤ sè ♥❣✉②➯♥ Z, Q, R, C ❧➔ ❝→❝ ♠✐➲♥ ♥❣✉②➯♥✳

❧➔ tr÷í♥❣ ♥➳✉ X ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ♠➔ tr♦♥❣ ✤â

♠å✐ ♣❤➛♥ tû ❦❤→❝ ❦❤æ♥❣ ✤➲✉ ❝â ♠ët ♥❣❤à❝❤ ✤↔♦ tr♦♥❣ ✈à ♥❤â♠ ♥❤➙♥
X✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ X

❱➼ ❞ö ✶✳✷✳✺✳
C✳

❚r÷í♥❣ sè ❤ú✉ t➾ Q✱ tr÷í♥❣ sè t❤ü❝ R✱ tr÷í♥❣ sè ♣❤ù❝

●✐↔ sû X ❧➔ ♠ët tr÷í♥❣✱ A ❧➔ ♠ët ❜ë ♣❤➟♥ ❝õ❛
X ê♥ ✤à♥❤ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr♦♥❣ X ✳ A ✤÷ñ❝ ❣å✐ ❧➔ ♠ët tr÷í♥❣ ❝♦♥
❝õ❛ tr÷í♥❣ X ♥➳✉ A ❝ò♥❣ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ ❝↔♠ s✐♥❤ tr➯♥ A ❧➔ ♠ët
tr÷í♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳

●✐↔ sû A ❧➔ ♠ët ❜ë ♣❤➟♥ ❝â ♥❤✐➲✉ ❤ì♥ ♠ët ♣❤➛♥ tû
❝õ❛ ♠ët tr÷í♥❣ X ✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✐✮ A ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ X ✳
✐✐✮ ❱î✐ ∀x, y ∈ A✱ x + y ∈ A, xy ∈ A✱ −x ∈ A, x−1 ∈ A ♥➳✉ x = 0✳
✐✐✐✮ ❱î✐ ∀x✱ y ∈ A, x − y ∈ A✱ xy−1 ∈ A✱ ♥➳✉ y = 0✳

✣à♥❤ ❧þ ✶✳✷✳✼✳

❚r÷í♥❣ ❝→❝ sè ❤ú✉ t➾ Q ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ tr÷í♥❣
❝→❝ sè t❤ü❝ R ✈➔ tr÷í♥❣ R ❧↕✐ ❧➔ tr÷í♥❣ ❝♦♥ ❝õ❛ tr÷í♥❣ ❝→❝ sè ♣❤ù❝ C✳
❱➼ ❞ö ✶✳✷✳✽✳

✶✵

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✶✳✸ ■✤➯❛♥ ✈➔ ✤ç♥❣ ❝➜✉ ✈➔♥❤
✶✳✸✳✶

■✤➯❛♥

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❈❤♦ X

❧➔ ♠ët ✈➔♥❤✱ I ❧➔ ✈➔♥❤ ❝♦♥ ❝õ❛ X ✳ ❑❤✐ ✤â
I ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ tr→✐ ✭♣❤↔✐✮ ♥➳✉ ∀x ∈ X, a ∈ I t❤➻ xa ∈ I ✭ax ∈ I ✮✳
I ✈ø❛ ❧➔ ✐✤➯❛♥ tr→✐ ✈ø❛ ❧➔ ✐✤➯❛♥ ♣❤↔✐ t❤➻ I ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ❝õ❛ X ✳
❱➼ ❞ö ✶✳✸✳✷✳
X✳

✲ ▼ët ✈➔♥❤ X ✤➲✉ ❝❤ù❛ ❤❛✐ ✐✤➯❛♥ t➛♠ t❤÷í♥❣ ④✵⑥ ✈➔

✲ ❚➟♣ ❤ñ♣ mZ ❣ç♠ ❝→❝ sè ♥❣✉②➯♥ ❧➔ ❜ë✐ ❝õ❛ ♠ët sè ♥❣✉②➯♥ m ❝❤♦
tr÷î❝ ❧➔ ♠ët ✐✤➯❛♥ tr♦♥❣ ✈➔♥❤ Z✳
✣➦❝ tr÷♥❣ ❝õ❛ ✐✤➯❛♥ ✤÷ñ❝ ❝❤♦ ❜ð✐ ♠➺♥ ✤➲ s❛✉✳
▼➺♥❤ ✤➲ ✶✳✸✳✸✳ ❈❤♦ X

❧➔ ♠ët ✈➔♥❤✱ I ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛
X ✳ ❑❤✐ ✤â ■ ❧➔ ✐✤➯❛♥ ❝õ❛ X ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ∀a, b ∈ I, ∀x ∈ X t❤➻





a−b∈I



ax ∈ I





xa ∈ I.

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✹✳ ❈❤♦ X
X✳

❧➔ ✈➔♥❤✱ I ❧➔ ✐✤➯❛♥ t❤ü❝ sü ❝õ❛ X ✱ ❦❤✐ ✤â✿
✐✮ I ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ♥➳✉ ❝❤➾ ❝â ❤❛✐ ✐✤➯❛♥ ❝❤ù❛ ♥â ❧➔ I ✈➔

✐✐✮ I ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➳✉ ∀x, y ∈ X ♠➔ xy ∈ I t❤➻ x ∈ I
❤♦➦❝ y ∈ I.
✐✐✐✮ I ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì ♥➳✉ ∀x, y ∈ X; xy ∈ I; y ∈/ I t❤➻
∃n ∈ N : xn ∈ I ✳
✶✶


õ tốt ồ

ũ

I ữủ ồ t q I = I1 I2 tr õ I1, I2
ừ X t I = I1 I = I2.
pZ ứ ỹ ứ sỡ
ụ tố ừ Z ợ p số tố
I = 7Z t q ừ Z
ử

X Y

f : X Y ởt ỗ
I ừ X f (I)Y ừ Y s f (I)
ữủ ồ rở ừ I tr Y
X Y

f : X Y ởt ỗ
J ừ Y t f 1(J) = {x X|f (x) J}
ừ X ữủ ồ rút ừ J tr X
ởt ợ rt q trồ ỳ s
X ởt A ởt t ừ X
ừ tt ừ X ự A ọ t ự A
ữủ ồ s A A õ A ữủ ồ t
s ừ A
t s A õ ỳ tỷ t ữủ ồ ỳ
s



X õ ỡ A ởt t
ừ X õ

ỵ

n

A ={

ai xi |n N; ai X; xi A, i = 1, n}.
i=1

ự t B = {

n
i=1 ai xi |n



N; ai X; xi A, i = 1, n}


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

●å✐ a = a1x1 + ... + anxn✱ b = a1y1 + ... + anyn ∈ B ✈➔ x ∈ X ✳
❉➵ t❤➜② a − b ∈ B ✈➔ xa = ax ∈ B ✳ ❙✉② r❛ B ❧➔ ✐✤➯❛♥ ❝õ❛ X ✳ ❱➻
∀x ∈ A : x = 1.x ♥➯♥ t❛ ❝â A ⊆ B ✳
❍ì♥ ♥ú❛✱ ✈î✐ ♠å✐ x1, ..., xn ∈ A ♠å✐ ✐✤➯❛♥ ❝õ❛ X ❝❤ù❛ A t❤➻ ✤➲✉
❝❤ù❛ a1x1, ..., anxn ♥➯♥ ❝ô♥❣ ❝❤ù❛ x1a1 + ... + xnan✳ ❱➟② B ❧➔ ✐✤➯❛♥
♥❤ä ♥❤➜t ❝❤ù❛ A✳ ❚❛ ♥❤➟♥ ✤÷ñ❝ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ ♠➺♥❤ ✤➲✳
❱➼ ❞ö ✶✳✸✳✶✵✳

■✤➯❛♥

❳➨t tr➯♥ ✈➔♥❤ sè ♥❣✉②➯♥ Z

3, 4, 5 = {3x + 4y + 5z|x, y, z ∈ Z} = Z.

✶✳✸✳✷

✣ç♥❣ ❝➜✉ ✈➔♥❤

❈❤♦ →♥❤ ①↕ f : X → Y ✈î✐ X, Y ❧➔ ❤❛✐ ✈➔♥❤✳
❑❤✐ ✤â f ❣å✐ ❧➔ ✤ç♥❣ ❝➜✉ ✈➔♥❤ ♥➳✉ t❤ä❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✐✮ f (a + b) = f (a) + f (b)✳
✐✐✮ f (ab) = f (a)f (b) ✈î✐ ∀a, b ∈ X ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✶✳

◆➳✉ ✤ç♥❣ ❝➜✉ ✈➔♥❤ f ❧➔ ✤ì♥ →♥❤ ✭t♦➔♥ →♥❤✱ s♦♥❣ →♥❤✮ t❤➻ f ✤÷ñ❝
❣å✐ ❧➔ ✤ì♥ ❝➜✉ ✭t♦➔♥ ❝➜✉✱ ✤➥♥❣ ❝➜✉✮✳
✰ ⑩♥❤ ①↕ f : X → Y ✱ x → 0 ❧➔ ✤ç♥❣ ❝➜✉✳
✰ ❈❤♦ S ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ X t❤➻ →♥❤ ①↕ i : a → a, ∀a ∈ S ❧➔
✤ì♥ ❝➜✉ ✭♣❤➨♣ ♥❤ó♥❣✮✳

❱➼ ❞ö ✶✳✸✳✶✷✳

❈❤♦ X ❧➔ ♠ët ✈➔♥❤✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ X ✳
❚➟♣ X/I = {x + I|x ∈ X} ❝ò♥❣ ✈î✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♥❤➙♥ ①→❝ ✤à♥❤
♥❤÷ s❛✉✿
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✸✳

(x + I) + (y + I) = (x + y) + I,

✶✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

(x + I).(y + I) = xy + I

✈î✐ ∀x, y ∈ X ❧➔ ♠ët ✈➔♥❤ ✈➔ ❣å✐ ❧➔ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛ X t❤❡♦ I ✳
❈❤♦ ✈➔♥❤ X ❧➔ ♠ët ✈➔♥❤ tò② þ✳ ❑❤✐ ✤â✿
X/{0} = {x + {0}|x ∈ X} X ✱
X/X = {x + X|x ∈ X} = {X} {0}✳

❱➼ ❞ö ✶✳✸✳✶✹✳

❚➼♥❤ ❝❤➜t ✶✳✸✳✶✺✳
•

•

❈❤♦ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ X ✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ X ✳

◆➳✉ J ❧➔ ✐✤➯❛♥ ❝õ❛ X s❛♦ ❝❤♦ J ⊇ I t❤➻ J/I ❧➔ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤
t❤÷ì♥❣ X/I ✈➔ ✈î✐ r ∈ R t❛ ❝â r + I ∈ J/I ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ r ∈ J ✳
▼é✐ ✐✤➯❛♥ B ❝õ❛ R/I ✤➲✉ ❝â ❞↕♥❣ K/I ✈î✐ K ❧➔ ✐✤➯❛♥ ❝õ❛ X
t❤ä❛ ♠➣♥ K ⊇ I ✳

• J1 ✱ J2

❧➔ ❝→❝ ✐✤➯❛♥ ❝õ❛ X s❛♦ ❝❤♦ J1, J2 ⊇ I ✳ ❚❛ ❝â J1/I ⊇ J2/I
❦❤✐ ✈➔ ❝❤➾ ❦❤✐ J1 ⊇ J2✳

✶✳✹ ❱➔♥❤ ✤❛ t❤ù❝
✶✳✹✳✶

❳➙② ❞ü♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥

❈❤♦ A ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✶✱ t❛ ✤➦t
P = {(a0 , a1 , ..., an , ...)|ai ∈ A, i = 0, 1, ...

✈➔ ai = 0 ❤➛✉ ❦❤➢♣ ♥ì✐}

❳➨t ✷ ♣❤➨♣ t♦→♥ tr➯♥ P ♥❤÷ s❛✉✿
(a0 , a1 , ..., an , ...) + (b0 , b1 , ..., bn , ...) := (a0 + b0 , a1 + b1 , ..., an + bn , ...)
(a0 , a1 , ..., an , ...).(b0 , b1 , ..., bn , ...) := (c0 , c1 , ..., cn , ...)

✶✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✈î✐ ck = i+j=k aibj , k = 0, 1, 2, ...
❑❤✐ ✤â✱ (P, +, .) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳
❳➨t q✉② t➢❝ f : A → P ✱ a → f (a) = (a, 0, ...)✳
❉➵ t❤➜②

f (a + b) = f (a) + f (b),
f (a.b) = f (a).f (b)
f (a) = f (b) ⇔ a = b

✈î✐ ♠å✐ a, b ∈ A✳ ❱➟② f ❧➔ ✤ì♥ ❝➜✉
❉♦ f ❧➔ ✤ì♥ ❝➜✉ ♥➯♥ t❛ ❝â t❤➸ ❝♦✐ A ❧➔ ✈➔♥❤ ❝♦♥ ❝õ❛ P ✱ ✈➔ t❛ ❝â
t❤➸ ✤ç♥❣ ♥❤➜t ♠é✐ ♣❤➛♥ tû a ∈ A ✈î✐ ↔♥❤ f (a) ❝õ❛ ♥â tr♦♥❣ P ✳ ❚r♦♥❣
P ✱ t❛ ❦➼ ❤✐➺✉
x0 : = (1, 0, ...)
x1 : = (0, 1, ...)
x2 : = (0, 0, 1, ...)
...

❱➻ ❝→❝ ♣❤➛♥ tû ❝õ❛ P ❧➔ ❝→❝ ❞➣② (a0, a1, ..., an, ...) tr♦♥❣ ✤â ai ∈ A
✈➔ ❜➡♥❣ 0 ❤➛✉ ❦❤➢♣ ♥ì✐ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ sû❛ n ❧➔ sè ❧î♥ ♥❤➜t ✤➸
an = 0 ✭ tù❝ ❧➔ an+1 = an+2 = ... = 0✮✳ ❑❤✐ ✤â✱ ❝→❝ ♣❤➛♥ tû tr♦♥❣ P
❝â t❤➸ ✈✐➳t✿
(a0 , a1 , ..., an , ...) = (a0 , 0, ...) + ... + (0, ..., 0, an , 0, ...)
= (a0 , 0, ...)(1, 0, ...) + ... + (an , 0, ...)(0, ..., 0, 1, 0, ...)
= a0 x0 + a1 x1 + ... + an xn ✳

✶✺


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

❑❤✐ ✤â✱ ✈➔♥❤ P ❝â t❤➸ ✈✐➳t
P = {a0 x0 + a1 x1 + ... + an xn |ai ∈ A, i = 0, n}.

❚❛ ❣å✐ P ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ ❝✉↔ ➞♥ x ❧➜② ❤➺ tû tr➯♥ ✈➔♥❤ A✱ ❦➼ ❤✐➺✉ A[x]✳
✭❳➙② ❞ü♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ♥❤✐➲✉ ❜✐➳♥✮
●✐↔ sû A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✱ t❛ ✤➦t✿

❈❤ó þ ✶✳✹✳✶✳

A1 = A[x1 ];
A2 = A[x2 ];
...
An = An−1 [xn ]

❱➔♥❤ An = An−1[xn]✱ ❦➼ ❤✐➺✉ A[x1, ..., xn]✱ ❣å✐ ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ ❝õ❛ n
➞♥ x1, x2, ..., xn ❧➜② ❤➺ tû tr♦♥❣ ✈➔♥❤ A✳ ▼é✐ ♣❤➛♥ tû ❝õ❛ An ❣å✐ ❧➔ ♠ët
✤❛ t❤ù❝ ❝õ❛ n ➞♥ x1, ..., xn ✈î✐ ❤➺ tû tr♦♥❣ ✈➔♥❤ A✱ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❧➔
f (x1 , ..., xn )✳
▼é✐ ✤❛ t❤ù❝ f (x1, ..., xn) ❝â t❤➸ ✈✐➳t ❞÷î✐ ❞↕♥❣ s❛✉✿
f (x1 , ..., xn ) = c1 xa111 ...xan1n + ... + cm xa1m1 ...xanmn

✈î✐ ci ∈ A✱ ai1, ..., ain, i = 1, m ❧➔ ♥❤ú♥❣ sè tü ♥❤✐➯♥ ✈➔ (ai1, ..., ain) =
(aj1 , ..., ajn ) ❦❤✐ i = j ✳
✶✳✹✳✷

❚➼♥❤ ❝❤➜t ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝

✣à♥❤ ❧þ ✶✳✹✳✷✳

♠✐➲♥ ♥❣✉②➯♥✳

◆➳✉ A ❧➔ ♠✐➲♥ ♥❣✉②➯♥ t❤➻ ✈➔♥❤ ✤❛ t❤ù❝ A[x] ❝ô♥❣ ❧➔

✶✻


õ tốt ồ

ũ

K ởt trữớ g(x) tự 0 ừ
K[x] õ ồ tự f K[x] õ t t ữợ

ỵ

f (x) = q(x)g(x) + r(x)

tr õ q(x), r(x) K[x] r(x) = 0 deg r(x) < deg f (x)
ỡ ỳ q(x) r(x) ữủ t

tr
õ ỡ X

ởt tr
ồ ừ õ ỳ s tỗ t ởt t s ỳ
tỷ

ồ ừ số Z õ nZ
ỳ s õ tr
rữớ số Q R C tr ồ trữớ õ
(0) (1)
ử

ởt õ ỡ õ
s tữỡ ữỡ
A tr
ộ t rộ ừ A ổ tỗ t ởt tỷ ỹ

ợ I1 I2 ... Ik Ik+1... ởt t ừ
A ổ tỗ t n In = In+1 = ... tự ồ t ừ A
ứ

ỵ A




õ tốt ồ
ỵ

tr

ũ

A ởt tr t tự A[x]

A tr t tự
A[x1 , ..., xn ] ụ tr

q

K ởt trữớ t K[x1, ..., xn] tr

ữỡ õ ừ ổ
t õ ỡ t
ổ tr õ ụ ổ
R õ ỡ ởt õ ở
M ữủ ồ Rổ tỗ t ởt ồ ổ
ữợ R ì M M (, x) x s s tọ
( + )x = x + x
(x + y) = x + y
()x = (x)
1x = x ợ ồ , R x, y M


t

tỡ tr R

R ởt trữớ t Rổ ởt ổ

M Rổ ồ X = {xi}iI ữủ ồ
s ừ M ồ tỷ ừ M õ t t ữợ
tờ ủ t t ừ tỷ mi ợ ồ m M tỗ t
ởt ồ tỷ ai R i I s m = iI aimi ổ M
ữủ ồ ởt ổ ỳ s M õ s ỳ

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

✤÷ñ❝ ❣å✐ ❧➔ ♠ët R✲♠æ✤✉♥ tr✉♥❣ t❤➔♥❤ ♥➳✉
❧✐♥❤ ❤â❛ tû ❝õ❛ ♥â ❜➡♥❣ 0✱ tù❝ ❧➔ AnnM (x) = {a ∈ M |ax = 0} ❜➡♥❣
0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✹✳ M

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✺✳ ❈❤♦ M

❧➔ R✲♠æ✤✉♥✱ S ❧➔ t➟♣ ❝♦♥ ♥❤➙♥ ✤â♥❣ ❝õ❛
R✳ ❚r➯♥ M × S = {(m, s)|m ∈ M, s ∈ S} t❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ ∼
♥❤÷ s❛✉✿ ✈î✐ ♠å✐ (m, s), (m , s ) ∈ M × S
(m, s) ∼ (m , s ) ⇔ ∃t ∈ S : t(ms − sm ) = 0.

❑❤✐ ✤â ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ ∼ ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥
M × S ✳ ❙✉② r❛ M × S ✤÷ñ❝ ❝❤✐❛ t❤➔♥❤ ❝→❝ ❧î♣ t÷ì♥❣ ✤÷ì♥❣✱ ❦➼ ❤✐➺✉
(m, s) ❧➔ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ❝❤ù❛ (m, s) tù❝ ❧➔
(m, s) = {(m , s ) ∈ M × S|(m, s) ∼ (m , s )}.

✣➸ t❤✉➟♥ t✐➺♥ t❛ ❦➼ ❤✐➺✉ ms t❤❛② ❝❤♦ (m, s)✱ ❦➼ ❤✐➺✉ S −1M ❧➔ t➟♣ t❤÷ì♥❣
❝õ❛ M × S t❤❡♦ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ∼✱ tù❝ ❧➔
S −1 M = {

m

|m ∈ M, s ∈ S}.
s

❑❤✐ M = R t❛ ❝â t➟♣ ❤ñ♣
r
S −1 R = { |r ∈ R, s ∈ S}.
s

✶✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

❚r➯♥ S −1R t❛ tr❛♥❣ ❜à ❤❛✐ ♣❤➨♣ t♦→♥ ✭✰✮✱ ✭✳✮ ♥❤÷ s❛✉✿
r r
s r + sr
+ =
s s
ss
rr
r r
. =
s s
ss

✈î✐ ♠å✐ rs , rs ∈ S −1R.
❱î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr➯♥ S −1R trð t❤➔♥❤ ♠ët ✈➔♥❤ ❣å✐ ❧➔ ✈➔♥❤ ❝→❝
t❤÷ì♥❣ ❝õ❛ R t❤❡♦ t➟♣ ♥❤➙♥ ✤â♥❣ S ❤❛② ❣å✐ ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ❤â❛

❝õ❛ R t❤❡♦ t➟♣ ♥❤➙♥ ✤â♥❣ S ✳
❚÷ì♥❣ tü ♥❤÷ ✈➟② tr➯♥ S −1M t❛ tr❛♥❣ ❜à ❤❛✐ ♣❤➨♣ t♦→♥ ✭✰✮✱ ✭✳✮
♥❤÷ s❛✉✿
m m
s m + sm
+
=
s
s
ss
r m rm
. =
t s
ts

✈î✐ ♠é✐ ms , ms ∈ S −1M ✱ rt ∈ S −1R✳
❱î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr➯♥ S −1M ❝ò♥❣ ✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr➯♥ trð t❤➔♥❤
♠ët S −1M ✲♠æ✤✉♥ ✤÷ñ❝ ❣å✐ ❧➔ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ♠æ✤✉♥ M t❤❡♦ t➟♣
♥❤➙♥ ✤â♥❣ S ✳
❈❤ó þ ✶✳✻✳✻✳

✐✮ ❚r♦♥❣ ✈➔♥❤ S −1R t❤➻
r
r
= ⇔ ∃t ∈ S : t(s r − sr ) = 0.
s s

✐✐✮ ❚r♦♥❣ ✈➔♥❤ S −1M t❤➻
m m
=

⇔ ∃t ∈ S : t(s m − sm ) = 0.
s
s

✷✵


õ tốt ồ

ũ

S 1M õ trú ởt Rổ ổ ữợ
r R,

m
m r m mr
S 1 M : r = . =
.
s
s
1 s
s

t S 1R ụ ởt Rổ
R ởt P ởt tố ừ R t S =
R\P = {x|x R, x
/ P } t õ ừ R
r
S 1 R = RP RP = { |r R, s S} õ ữỡ ữủ
s

ồ ữỡ õ ừ R t P ợ ỹ t
s
P RP = { |r P, s S} tr R RP ồ ữỡ õ
r
ỵ ỗ tỹ f : R S 1 R I

ừ
a
R t S 1 I = I e = f (I)S 1 R õ S 1 I = { |a I, s S}
s
1
ừ S R
ự S 1I s f (I) tr S 1R ồ
tỷ ừ S 1I õ
ri
ai
n a i ri
. ợ
S 1 R
f (I)
i=1
1 s
s
1

a1 r1 ia2 r2 i
an rn
=
+
+ ... +

s
s
s
a1 r1 1 s2 ...sn2
an rnns1 ...sn1
=
+ ... +
s1 ...sn
s1 ...sn
a
=
s
tr õ s = s1...sn S a = a1r1s2...sn + ... + anrns1...sn1 R

S 1I ừ S 1R

ỵ ỗ tỹ f : R S 1 R ồ ừ

õ S 1I ợ I ừ R ỡ ỳ tỹ sỹ
I S =

S 1 R




❈❤÷ì♥❣ ✷
❇❆❖ ✣➶◆● ◆●❯❨➊◆ ❈Õ❆
■✣➊❆◆
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❡ ♠✉è♥ ❣✐î✐ t❤✐➺✉ ✈➲ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥✱ ✤➛✉ t✐➯♥

❡♠ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✱ ♠ö❝ t✐➳♣ t❤❡♦ ❧➔ ♠æ t↔ ❜❛♦ ✤â♥❣
♥❣✉②➯♥ ✈î✐ r❡❞✉❝t✐♦♥ ✈➔ ❝✉è✐ ❝ò♥❣ ❧➔ ♣❤➛♥ ♥ë✐ ❞✉♥❣ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥
❝õ❛ ♠ët ✐✤➯❛♥✳
❚➜t ❝↔ ❝→❝ ✈➔♥❤ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❝❤÷ì♥❣ ✷ ✈➔ ❝❤÷ì♥❣ ✸ ✤➲✉ ❧➔ ✈➔♥❤
❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à✳

✷✳✶ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥
❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ P❤➛♥ tû r ∈ R
✤÷ñ❝ ❣å✐ ❧➔ ♥❣✉②➯♥ tr➯♥ I ♥➳✉ tç♥ t↕✐ ♠ët sè ♥❣✉②➯♥ n ✈➔ ❝→❝ ♣❤➛♥
tû ai ∈ I i ✱ i = 1, ..., n s❛♦ ❝❤♦
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳

rn + a1 rn−1 + a2 rn−2 + ... + an−1 r + an = 0✳

P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ♣❤ö t❤✉ë❝ ♥❣✉②➯♥
❝õ❛ r tr➯♥ I ✭❜➟❝ n✮✳
✷✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

◆❣✉②➵♥ ❚❤à ❚❤ò② ▲✐♥❤

❚➟♣ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❝õ❛ R ♥❣✉②➯♥ tr➯♥ I ✤÷ñ❝ ❣å✐ ❧➔ ❜❛♦ ✤â♥❣
♥❣✉②➯♥ ❝õ❛ I ✈➔ ❦➼ ❤✐➺✉ I ✳ ✣➦❝ ❜✐➺t✿
•

◆➳✉ I = I t❤➻ I ✤÷ñ❝ ❣å✐ ❧➔ ✤â♥❣ ♥❣✉②➯♥✳

•

◆➳✉ I ⊆ J ❧➔ ❝→❝ ✐✤➯❛♥ t❛ ♥â✐ J ❧➔ ♥❣✉②➯♥ I ♥➳✉ J ⊆ I ✳

•

◆➳✉ I ❧➔ ♠ët ✐✤➯❛♥ s❛♦ ❝❤♦ ✈î✐ ♠å✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n ♠➔ I n ❧➔
✤â♥❣ ♥❣✉②➯♥ t❤➻ I ❧➔ ✐✤➯❛♥ ❝❤✉➞♥ t➢❝✳

❱➼ ❞ö ✷✳✶✳✷✳ ❛✮ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✱ ❤❛✐ ♣❤➛♥ tû ❜➜t ❦➻ x, y ∈ R✳ ❑❤✐

✤â xy ❧➔ ♣❤➛♥ tû ♥❣✉②➯♥ ❝õ❛ ✐✤➯❛♥ I = (x2, y2)✱ tù❝ ❧➔ xy ∈ (x2, y2)✳
❇ð✐ ✈➻ t❛ ❝â
(xy)2 + a1 (xy) + a2 = 0

ð ✤â a1 = 0 ∈ I ✱ a2 = −x2y2 ∈ I 2✳
❜✮ ❚÷ì♥❣ tü✱ ❝❤♦ d > 0 t❛ ❝â xiyd−i ❧➔ ❝→❝ ♣❤➛♥ tû ♥❣✉②➯♥ tr➯♥
✐✤➯❛♥ (xd, yd)✱ ∀i d✳
❉÷î✐ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥✳
✶✳ I ⊆ I ✳
✷✳ ◆➳✉ I ⊆ J ❧➔ ❝→❝ ✐✤➯❛♥ t❤➻ I ⊆ J ✱ tù❝ ❧➔ ♠é✐ ♣❤➛♥ tû ♥❣✉②➯♥
tr➯♥ I ❧➔ ♥❣✉②➯♥ tr➯♥ J ✳
√
✸✳ I ⊆ I ✳
✹✳ ❈→❝ ✐✤➯❛♥ ❝➠♥✱ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧➔ ✤â♥❣ ♥❣✉②➯♥✳
√
√
✺✳ 0 ⊆ I ✈î✐ ♠å✐ ✐✤➯❛♥ I ✱ ð ✤â 0 = {r ∈ R|∃n > 0 s❛♦ ❝❤♦
rn = 0}✲❝➠♥ ❧ô② ❧✐♥❤ ❝õ❛ ✈➔♥❤✳
✻✳ ●✐❛♦ ❝õ❛ ❝→❝ ✤â♥❣ ♥❣✉②➯♥ ❧➔ ✤â♥❣ ♥❣✉②➯♥✳
✼✳ ◆➳✉ ϕ : R → S ❧➔ ✤ç♥❣ ❝➜✉ ✈➔♥❤ t❤➻ ϕ(I) ⊆ ϕ(I)S ✳

▼➺♥❤ ✤➲ ✷✳✶✳✸✳

✷✸