❚❘×❮◆● ✣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❝ ❉✐➺♣ ỡ rữợ tr ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ tợ t ổ trữớ ữ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ❜ë ♠ỉ♥ ✤↕✐ sè ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ tr✐ t❤ù❝ qỵ t t ủ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ö ❦❤â❛ ❤å❝ ✈➔ ❦❤â❛ ❧✉➟♥✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❣✐→♦ ✲ ❚❤↕❝ s ộ ữớ trỹ t ữợ ❞➝♥✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥✱ ♥➠♥❣ ❧ü❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜↔♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ rt ữủ ỳ ỵ õ ỵ qỵ t ổ ❜↕♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ▲í✐ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✧■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❞♦ sü ❝è ❣➢♥❣✱ ♥é ❧ü❝ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈ỵ✐ 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❝ ✈➔ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ♥➔♦ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✶ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✹ ✶✳✶ ❱➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❱➔♥❤ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ■✤➯❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ ❱➔♥❤ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ỗ ✳ ✳ ✳ ✳ ✶✵ ✶✳✻ ❱➔♥❤ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✼ ▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✽ ▼æ✤✉♥ ❝♦♥ ✶✺ ✶✳✾ ▼æ✤✉♥ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ổ ỗ ❝➜✉ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✶✶ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ✈➔♥❤ ✈➔ ♠ỉ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷ ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚ ✷✹ ✸ ❙Ü P❍❹◆ ❚➑❈❍ ◆●❯❨➊◆ ❙❒ ✸✾ ❑➳t ❧✉➟♥ ✹✼ ✶ ❑❤â❛ ❧✉➟♥ tèt P é ỵ ❞♦ ❝❤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 ✈ỵ✐ ♠ët ♠ỉ✤✉♥✳ ◆❣♦➔✐ r❛ ỉ♥❣ ❝á♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤à♥❤ ❧➼ ❧å❝ ❇♦✉r❜❛❦✐ ✈➔ ❤➺ q✉↔ ❝õ❛ ♥â ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝ơ♥❣ ❧➔ ♠ët ✤è✐ t÷đ♥❣ 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➯♥ ✈➔ ♠ỉ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② t❤➻ ♥❣✉②➯♥ sì✳ ◆❤ú♥❣ ✈➜♥ ✤➲ tr➯♥ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✤↕✐ sè ✈➔ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ 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 ❝ơ♥❣ s➩ ✤÷❛ r❛ ❤➺ t❤è♥❣ ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ ❜ê ✤➲✱ ♥❤➟♥ ①➨t ✤➸ ✤÷❛ ✤➳♥ ❝→❝ ❦➳t q✉↔ ♥➯✉ tr➯♥✳ ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✈➔♥❤✱ ✈➔♥❤ ◆♦❡t❤❡r✱ ✐✤➯❛♥✱ ♠ỉ✤✉♥✱ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ✈➔♥❤ ✈➔ ♠ỉ✤✉♥✱ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â✱ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ❣✐→♦ tr➻♥❤✱ s→❝❤ t❤❛♠ ❦❤↔♦ ✈➔ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉✳ ✺✳ ❈➜✉ trú õ ỗ ữỡ ữỡ 1✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❝õ❛ ✤↕✐ sè ✤â ❧➔✿ tữỡ tr ỗ ✈➔♥❤✱ ♠ỉ✤✉♥✱ ♠ỉ✤✉♥ ❝♦♥✱ ♠ỉ✤✉♥ t❤÷ì♥❣✱ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ♠ỉ✤✉♥✱ ❞➣② ❦❤ỵ♣✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ♣❤ư❝ ✈ư ❝❤♦ ự ỵ ữỡ s ữỡ tố ❦➳t ❈❤÷ì♥❣ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ✈➲ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝ò♥❣ ❝→❝ ♠➺♥❤ ✤➲✱ ✤à♥❤ ❧➼ ❧✐➯♥ q✉❛♥ ✈➲ t➟♣ ❝→✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ ◆❣♦➔✐ r tr ữỡ ỏ ợ t ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼ ❧å❝ ❇♦✉r❜❛❦✐ tø ✤â ✤÷❛ r❛ ❦➳t ❧✉➟♥ ✈➲ t➼♥❤ ❤ú✉ ❤↕♥ ❝õ❛ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r✳ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ • ❈❤÷ì♥❣ 3✿ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❜➔② ✈➲ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ P❤➛♥ ✤➛✉ ỗ õ ổ sỡ t ♥❣✉②➯♥ ✈➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉②✳ ▼ët ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ✈➲ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ s➩ ✤÷đ❝ ✤÷❛ r❛ tr♦♥❣ ♣❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣✳ ✸ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❝õ❛ ✤↕✐ sè ❣✐❛♦ ❤♦→♥✱ ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ P❤➛♥ t ữỡ ỗ ởt số tự P tự ữỡ ỗ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♠ỉ✤✉♥✱ ♠ỉ✤✉♥ ❝♦♥✱ ♠ỉ✤✉♥ t❤÷ì♥❣✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥✳ ▼ët sè ❦❤→✐ ♥✐➺♥ ✈➲ ữỡ õ ổ ợ s ữủ ð ♣❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣✳ ✶✳✶ ❱➔♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ R ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✳ ❑❤✐ ✤â R ❝ò♥❣ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♥❤➙♥✱ ❦➼ ❤✐➺✉ ❧➔ (+) ✈➔ (.) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ R ❝ò♥❣ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ❧➔ ♠ët ♥❤â♠ ❆❜❡❧❀ ✭✐✐✮ R ❝ò♥❣ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ❧➔ ♠ët ♥û❛ ♥❤â♠❀ ✭✐✐✐✮ P❤➨♣ ♥❤➙♥ ♣❤➙♥ ♣❤è✐ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ x, y, z ∈ R ✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ t❤➻ (x + y)z = xz + yz ✈➔ z(x + y) = zx + zy ✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❝â ✤ì♥ ✈à ♥➳✉ R ❧➔ ♠ët ✈à ♥❤â♠ ♥❤➙♥✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ♥➳✉ ♣❤➨♣ ♥❤➙♥ ❝â t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ♥➳✉ R ❧➔ ♠ët ✈à ♥❤â♠ ♥❤➙♥ ❣✐❛♦ ❤♦→♥✳ ❱➼ ❞ö ✶✳✷✳ Z, Q, R, Z[x] ú ỵ ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔② ♣❤➛♥ tû ❦❤æ♥❣ ❝õ❛ ổ ữủ ỵ P tỷ ỡ õ ổ ữủ ỵ 1✳ ▼➺♥❤ ✤➲ s❛✉ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ tr♦♥❣ ✈➔♥❤✳ ▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✳ ❑❤✐ ✤â ✶✳ 0x = x0 = ✈ỵ✐ ♠å✐ x ∈ R❀ ✷✳ (−x)y = x(−y) = −(xy) ✈ỵ✐ ♠å✐ x, y ∈ R❀ ✸✳ (−x)(−y) = xy ✈ỵ✐ ♠å✐ x, y ∈ R❀ ✹✳ x(y − z) = xy − xz; (x − y)z = xz − yz ✈ỵ✐ ♠å✐ x, y, z ∈ R❀ ✺✳ (−x)2n = x2n ; (−x)2n+1 = −x2n+1 ✈ỵ✐ ♠å✐ x R, n N ìợ ❝õ❛ ❦❤æ♥❣✮ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✳ ❚❛ ❣å✐ ♣❤➛♥ tỷ = a R ữợ tỗ t = b R tọ q✉❛♥ ❤➺ ab = 0✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ✭▼✐➲♥ ♥❣✉②➯♥✮ ▼ët ✈➔♥❤ ❝â ♥❤✐➲✉ ❤ì♥ ✶ ♣❤➛♥ tû✱ ❣✐❛♦ ❤♦→♥✱ õ ỡ ổ õ ữợ ữủ ❧➔ ♠✐➲♥ ♥❣✉②➯♥✳ ✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ M = (0) t❤➻ AssR M = ∅ ✲ ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ◆➳✉ M = (0) t❤➻ t❤❡♦ ✣à♥❤ ❧➼ ❧å❝ ❇♦✉r❜❛❦✐ tr➯♥ M ❝â ❧å❝ (0) = Mn ··· Mn−1 M1 M0 = M ❚❛ ❝â ❝→❝ ❞➣② ❦❤ỵ♣ −→ Mn −→ Mn−1 −→ Mn−1 /Mn −→ ▼➔ Mn−1 /Mn ∼ = R/Qn ✈ỵ✐ ♠å✐ n ♥➯♥ t❛ ❝â ❞➣② ❦❤ỵ♣ −→ Mn −→ Mn−1 −→ R/Qn −→ 0 −→ Mn−1 −→ Mn−2 −→ R/Qn−1 −→ ··· −→ M1 −→ M0 −→ R/Q1 −→ ❑❤✐ ✤â t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✽ t❛ ❝â AssR Mn−1 ⊆ {Qn } AssR Mn−2 ⊆ {Qn , Qn−1 } ··· AssR M0 ⊆ {Q1 } ∪ {Qn−1 , · · · , Q2 } = {Q1 , Q2 , · · · , Qn } ▼➔ M0 = M ♥➯♥ AssR M ỳ ỵ M R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â AssR HomR (M, N ) = SuppR M ∩ AssR N ✈ỵ✐ ♠å✐ R✲♠ỉ✤✉♥ N ✳ ✸✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱ ❣✐↔ sû ❜ð✐ x1 , x2 , · · · , xn ✳ ❑❤✐ ✤â M = Rx1 + Rx2 + · · · + Rxn ✳ ❚❛ ❝â t♦➔♥ ❝➜✉ Rn −→ M −→ (a1 , a2 , · · · , an ) −→ a1 x1 + a2 x2 + · · · + an xn ❚→❝ ✤ë♥❣ Hom(❴, N ) ✈➔♦ ❞➣② tr➯♥ n −→ HomR (M, N ) −→ HomR (Rn , N ) ∼ = Nn = (HomR (R, N )) ∼ ❑❤✐ ✤â AssR HomR (M, N ) ⊆ AssR N n = AssR N = AssR N ✳ ❱ỵ✐ ♠å✐ Q ∈ AssR HomR (M, N ) t❛ ❝❤ù♥❣ ♠✐♥❤ Q ∈ SuppR M tù❝ ❧➔ ❝❤ù♥❣ ♠✐♥❤ MQ = ♠➔ MQ ∼ = RQ ⊗ M ♥➯♥ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ RQ ⊗ M = 0✳ ❱➻ Q AssR HomR (M, N ) tỗ t = f ∈ HomR (M, N ) s❛♦ ❝❤♦ Q = :R f ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ RQ ⊗ f = 0✳ ●✐↔ sû RQ ⊗R f = t❤➻ t❛ ❝â ⊗ f (xi ) = ✈ỵ✐ ♠å✐ i n✳ f (xi ) = ợ i n õ tỗ t↕✐ si ∈ / Q s❛♦ ❝❤♦ ❙✉② r❛ si f (xi ) = ✈ỵ✐ ♠å✐ i n✳ ✣➦t s = s1 s2 · · · sn t❤➻ s ∈ / Q ✈➔ sf (xi ) = ✈ỵ✐ ♠å✐ i n ❙✉② r❛ sf (x) = ✈ỵ✐ ♠å✐ x ∈ M, ❞♦ ✤â sf = 0✳ ❑❤✐ ✤â s ∈ :R f = Q ✭♠➙✉ t❤✉➝♥✮✳ ❉♦ ✤â RQ ⊗R f = 0✱ s✉② r❛ RQ ⊗ HomR (M, N ) = 0✳ ▼➔ RQ ⊗ HomR (M, N ) ∼ = HomRQ (MQ , NQ ) = ♥➯♥ t❛ ❝â MQ = (0)✳ ❚ø ✤â s✉② r❛ Q ∈ SuppR M ✳ ❉♦ ✤â AssR HomR (M, N ) ⊆ SuppR M ✳ ❱➟② AssR HomR (M, N ) ⊆ SuppR M ∩ AssR N ✳ ✸✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû Q ∈ SuppR M ∩ AssR N tù❝ ❧➔ Q ∈ SuppR M ✈➔ Q ∈ AssR N ✳ ❱➻ Q ∈ SuppR M ♥➯♥ t❛ ❝â MQ = (0) ✈➔ MQ ❧➔ RQ ✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❜ð✐ x1 x2 xn x1 x2 xn , , · · · , ✳ ❑❤✐ ✤â MQ = RQ + RQ + · · · + RQ ✳ 1 1 1 õ ỗ tỹ h : M −→ MQ ✈➔ ϕ : MQ −→ RQ /QRQ a1 x a2 x an xn / Q) + ··· + −→ + QRQ , (ai ∈ R, si ∈ s1 s2 sn si ❧➔ t♦➔♥ ❝➜✉✳ ✣➦t g = ϕ ◦ h : M −→ MQ −→ RQ /QRQ ❧➔ →♥❤ ①↕ ❤ñ♣ t❤➔♥❤✳ ❚❛ ❝â Im g ⊆ RQ /QRQ = bi + QRQ | bi ∈ RQ , i = 1, n tr♦♥❣ ✤â bi = ci , ci ∈ R, ti ∈ / Q✳ ✣➦t t = t1 t2 · · · tn ∈ / Q✳ ❑❤✐ ✤â t Im g ⊆ R/Q✳ ti ❚❤❛② g ❜ð✐ tg t❛ ❝â t❤➸ ❣✐↔ sû Im g ⊆ R/Q✳ ❑❤✐ ✤â g ❝❤♦ t❛ ♠ët →♥❤ ①↕ g : M −→ R/Q x −→ g(x) ✸✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ Q AssR N tỗ t ✤ì♥ ❝➜✉ ψ : R/Q −→ N ✳ ✣➦t g ψ λ = ψ ◦ g : M −→ R/Q −→ N x −→ g(x) ❑❤✐ ✤â :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 ✈ỵ✐ ♠å✐ P ∈ SuppR M t❤➻ P ∈ Var(I) ♥➯♥ P ⊇ I ✳ ❱➻ P ∈ SuppR M ♥➯♥ t❤❡♦ ▼➺♥❤ t tỗ t Q AssR M s ❝❤♦ P ⊇ Q✳ ❉♦ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛ √ I t❤➻ √ I ⊇ Q✳ ▼➦t ❦❤→❝ Q = AssR M tỗ t = x M ✤➸ Q = annR x ⊇ √ I ✳ ❱➟② Q = I ✳ √ ❱ỵ✐ a, b ∈ R, ab ∈ / I, b ∈ I t❛ ❝❤ù♥❣ ♠✐♥❤ a ∈ I ✳ annR M = I ❞♦ ✈➟② Q ⊇ I ⇒ Q ⊇ √ ❱➻ ab ∈ I ♥➯♥ abM = 0✳ ▼➔ bM = ✭❞♦ b / I s r a ữợ tr bM ụ ữợ tr➯♥ M ✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻ t❤➻ P ♠➔ AssR M = {Q} ♥➯♥ a ∈ Q = ZDR (M ) = √ I ✳ ❉♦ ✤â I ❧➔ P ∈AssR M ✐✤➯❛♥ ♥❣✉②➯♥ sì✳ ❱➟② I ❧➔ Q✲♥❣✉②➯♥ sỡ ỵ R tr M ❧➔ R✲♠æ✤✉♥ ❦❤→❝ (0)✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ✭✶✮ ❚➟♣ AssR M ❝â ❞✉② ♥❤➜t ✶ ♣❤➛♥ tû✳ ✸✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✭✷✮ ❱ỵ✐ ♠å✐ a ∈ R✱ →♥❤ ①↕ ♥❤➙♥ a ˆ : M −→ M x −→ ax ❧➔ ✤ì♥ ❝➜✉ ❤♦➦❝ ❧ơ② ❧✐♥❤ ✤à❛ ♣❤÷ì♥❣ ✭tù❝ ❧➔ ợ ộ x M tỗ t n N s❛♦ ❝❤♦ an x = 0✮✳ ❈❤ù♥❣ ♠✐♥❤✳ (1) ⇒ (2)✳ ●✐↔ sû AssR M ❝â ❞✉② ♥❤➜t ✶ ♣❤➛♥ tû✳ ✣➦t AssR M = {Q}✳ ❱ỵ✐ ❜➜t ❦➻ a ∈ R✱ ♥➳✉ a ∈ / Q t❤➻ t❤❡♦ ❍➺ q t a ổ ữợ tr M ✳ ❉♦ ✤â a ˆ ❧➔ ✤ì♥ ❝➜✉✳ ◆➳✉ a ∈ Q✱ t❛ ❝❤ù♥❣ ♠✐♥❤ a ˆ ❧➔ ❧ô② ữỡ t ợ = 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✳ √ ❉♦ õ a I tỗ t n > s❛♦ ❝❤♦ an ∈ I ♥➯♥ an x = 0✳ (2) ⇒ (1)✳ ❱➻ M = ♥➯♥ AssR M = ∅✳ ❱ỵ✐ P, Q ∈ AssR M t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ P = Q✳ ❚❤➟t ✈➟②✳ ❱➻ Q ∈ AssR M tỗ t = x M s❛♦ ❝❤♦ Q = (0) :R x✳ ❱ỵ✐ ♠å✐ a ∈ P t❛ ❝❤ù♥❣ ♠✐♥❤ a ∈ Q✳ ❱➻ P AssR M a ữợ tr M ❞♦ ✤â a ˆ ❦❤ỉ♥❣ t❤➸ ❧➔ ✤ì♥ ❝➜✉✳ ❉♦ ✈➟② →♥❤ ①↕ ♥❤➙♥ a ˆ ❧➔ ❧ô② ❧✐♥❤ tỗ t n > an x = 0✳ ❙✉② r❛ an ⊆ :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✳ ✸✽ ❈❤÷ì♥❣ ✸ ❙Ü P❍❹◆ ❚➑❈❍ ◆●❯❨➊◆ ❙❒ ◆ë✐ ❞✉♥❣ ♣❤➛♥ ❝✉è✐ ❝õ❛ ữỡ ợ t sỹ sỡ tr ✈➔♥❤✳ P❤➛♥ ✤➛✉ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ♥➯✉ ❧➯♥ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ✈➔ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ ❦❤✐ ✤à♥❤ ♥❣❤➽❛ ♥â✳ P❤➛♥ t✐➳♣ t❤❡♦ ❝â ♥ë✐ ❞✉♥❣ ỗ q t q✉② ❝õ❛ ♠ët ♠æ✤✉♥ ❝♦♥✱ tø ✤â t✐➳♣ ❝➟♥ ✤à♥❤ ♥❣❤➽❛ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✈➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉②✳ ▼ët ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ✈➲ ♠è✐ q✉❛♥ ❤➺ ❜➜t ❦❤↔ q✉② ✈➔ ♥❣✉②➯♥ ✤÷đ❝ ♥➯✉ r❛ ð ♣❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✳ M ❧➔ R✲♠æ✤✉♥✳ N ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❚❛ ♥â✐ N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ♥➳✉ ✈ỵ✐ ♠å✐ x ∈ M \ N, a R tọ ax N t tỗ t↕✐ n ∈ N∗ s❛♦ ❝❤♦ an M ⊆ N ✳ ▼➺♥❤ ✤➲ ✸✳✷✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥✳ N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ✳ ❑❤✐ ✤â ❤❛✐ s tữỡ ữỡ ợ x M \ N, a ∈ R t❤ä❛ ♠➣♥ ax ∈ N t tỗ t n N s an M ⊆ N ✳ ✭✷✮ ◆➳✉ a ∈ R ❧➔ ÷ỵ❝ ❝õ❛ tr➯♥ M/N t❤➻ a ∈ ✸✾ annR (M/N )✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ❈❤ù♥❣ ♠✐♥❤✳ (1) ⇒ (2)✳ ●✐↔ sû ❝â (1) a ữợ tr M/N õ tỗ t = x + N M/N s❛♦ ❝❤♦ a(x + N ) = N t❤➻ t❛ ❝â x∈ / N ✈➔ ax ∈ N ✳ ❚❤❡♦ (1) t❛ s✉② r❛ ∃n ∈ N∗ s❛♦ ❝❤♦ an M ⊆ N ✱ s✉② r❛ an M/N = tr♦♥❣ M/N ✳ ❉♦ ✤â a ∈ annR (M/N )✳ (2) ⇒ (1)✳ ●✐↔ sû ❝â (2)✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x ∈ M \ N, a ∈ N t❤ä❛ ♠➣♥ ax ∈ N t❛ ❝â x + N = tr♦♥❣ M/N ✭❞♦ x ∈ / N ✮✳ ▼➔ a(x + N ) = ax + N = tr M/N s r a ữợ tr♦♥❣ M/N ✳ ❚❤❡♦ (2) t❤➻ a ∈ annR (M/N )✱ tù❝ ❧➔ ∃n ∈ N∗ s❛♦ ❝❤♦ an M/N = 0✳ ❉♦ ✤â an M ⊆ N ✳ ✣à♥❤ ỵ R tr M Rổ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â N ⊆ M ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t➟♣ AssR M/N ❝❤➾ ❝â ♠ët ♣❤➛♥ tû✱ tù❝ ❧➔ |AssR M/N | = 1✳ ❍ì♥ ♥ú❛✱ ♥➳✉ AssR M/N = {P } √ ✈➔ annR (M/N ) = I t❤➻ I ❧➔ P ♥❣✉②➯♥ ✈➔ I = P ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû AssR M/N = {P }✳ ❚❛ ❝â SuppR M/N = Var(annR (M/N )) = Var(P )✳ ❉♦ ✤â P = annR (M/N ) ✭✈➻ Var(I) = Var(J) ⇔ √ I= √ J ✮✳ ●✐↔ sû a ❧➔ ÷ỵ❝ ❝õ❛ tr♦♥❣ M/N ✱ ❦❤✐ ✤â a ∈ ZDR (M/N ) = Q = {P } ⇒ a ∈ P ⇒ a ∈ annR (M/N )✳ Q∈AssR M ❱➟② N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M t❤➻ N = M ✱ ❞♦ ✤â AssR M/N = ∅✳ ✣➦t I = annR (M/N )✳ ❚❛ s➩ ❝❤➾ r❛ I ❧➔ P ✲♥❣✉②➯♥ ✭ð ✤â √ I = P ✈ỵ✐ P ∈ Spec R✮ ✈➔ AssR M/N = {P }✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ Q ∈ AssR M/N t❤➻ t❛ ❝â ✈ỵ✐ a ∈ Q t❤➻ a ∈ ZDR (M/N ) ♥➯♥ s✉② r❛ a ∈ annR (M/N ) = √ I ✳ ❉♦ ✤â Q ⊆ ✹✵ √ I✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ▼➦t ❦❤→❝✱ Q ∈ AssR M/N ♥➯♥ tỗ t = x M/N tọ Q = annR x¯ ⊇ annR (M/N ) = I ✳ ❉♦ ✤â Q ⊇ √ ❱➟② Q = I ✳ ❑❤✐ ✤â |AssR M/N | = 1✳ √ I✳ ❈✉è✐ ❝ò♥❣ t❛ ❝➛♥ ❝❤➾ r❛ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì✳ ❚❤➟t ✈➟② ✈ỵ✐ ♠å✐ a, b ∈ R, ab ∈ I, b ∈ / I t❤➻ t❛ ❝â ab(M/N ) = ✈➔ b(M/N ) = 0✳ ❑❤✐ ✤â √ a ∈ ZDR (M/N ) ♥➯♥ s✉② r❛ a ∈ annR (M/N ) = I ✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✹✳ ❈❤♦ M ❧➔ R✲♠ỉ✤✉♥✱ N ❧➔ R✲♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ✳ ◆➳✉ AssR M/N = {P } ð ✤â P = annR (M/N ) t❤➻ t❛ ♥â✐ r➡♥❣ N ❧➔ ♠ỉ✤✉♥ ❝♦♥ P ✲♥❣✉②➯♥ ❝õ❛ M ✳ ỵ N N ổ ❝♦♥ P ✲♥❣✉②➯♥ ❝õ❛ M t❤➻ N ∩ N ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ P ✲♥❣✉②➯♥ sì✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✤ì♥ ❝➜✉ M/(N ∩ N ) −→ M/N ⊕ M/N x + N ∩ N −→ (x + N, x + N ) ❙✉② r❛ ∅ = AssR M/(N ∩ N ) ⊆ AssR M/N ∪ AssR M/N = {P } ❙✉② r❛ AssR M/(N ∩ N ) = {P } ❚❤❡♦ ✣à♥❤ ❧➼ ✸✳✸ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✻✳ ❈❤♦ N ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❚❛ ♥â✐ N ❧➔ ❦❤↔ q✉② ♥➳✉ N = N1 ∩ N2 tr♦♥❣ ✤â N1 , N2 ❧➔ ✷ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ 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 ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❜➜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 ✈ỵ✐ N1 , N2 = N ❞♦ ✤â N N1 , N2 ✳ ❉♦ t➼♥❤ ❝ü❝ ✤↕✐ ❝õ❛ N0 s✉② r❛ N1 , N2 ∈ / ❚✲ ♥❣❤➽❛ ❧➔ N1 , N2 ✤➲✉ ✈✐➳t ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❤ú✉ ❤↕♥ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ♥➯♥ N0 ❝ơ♥❣ ✈➙② t = ú ỵ ❙ü ❜✐➸✉ ❞✐➵♥ ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❧➔ ❦❤ỉ♥❣ ❞✉② ♥❤➜t✳ ❱➼ ❞ư ✸✳✶✶✳ ❈❤♦ R ❧➔ ♠ët tr÷í♥❣ ✈➔ M ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì n ❝❤✐➲✉ tr➯♥ R✳ ❑❤✐ ✤â ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❝õ❛ M ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✹✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✈➨❝tì ❝♦♥ n − ❝❤✐➲✉ ✳ ❘ã r➔♥❣ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈➨❝tì tì n − ❝❤✐➲✉ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ n − ❝❤✐➲✉ t❤❡♦ ♥❤✐➲✉ ❝→❝❤ ❦❤→❝ ♥❤❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✷✳ ❚➟♣ N ❝â ❜✐➸✉ ❞✐➵♥ N = N1 ∩ N2 · · · ∩ Nr (∗) tr♦♥❣ ✤â N = Nj ✈ỵ✐ ♠å✐ i = 1, r✳ ◆➳✉ Ni ❧➔ ❜➜t ❦❤↔ q✉② ✭♥❣✉②➯♥ j=i,i=1,r sì✮ t❤➻ (∗) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉② t❤✉ ❣å♥ ✭♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥✱ t÷ì♥❣ ù♥❣✮ ❝õ❛ N ✳ ◆❤➟♥ ①➨t ✸✳✶✸✳ ❈❤♦ N = N1 ∩ N2 · · · ∩ Nr ❧➔ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✈ỵ✐ AssR M/Ni = {Pi } ✈ỵ✐ i = 1, r ✳ ◆➳✉ Pi = Pj t❤➻ t❤❡♦ ✤à♥❤ ❧➼ ✸✳✺ Ni ∩ Nj ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ Pi ✲♥❣✉②➯♥ sì✳ ❇ð✐ ✈➟② ♥❤â♠ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ t÷ì♥❣ ù♥❣ ❝ò♥❣ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè t❛ ♥❤➟♥ ✤÷đ❝ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ♠➔ Pi = Pj ✈ỵ✐ ♠å✐ i = j ✱ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ♥❣➢♥ ♥❤➜t ✭❝ơ♥❣ ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✮✳ ❇ê ✤➲ ✸✳✶✹✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ◆➳✉ N ♥❣✉②➯♥ tr♦♥❣ M t❤➻ ♠ỉ✤✉♥ ❝♦♥ (0) ❧➔ ♥❣✉②➯♥ tr♦♥❣ M/N ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ N ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ tr♦♥❣ M ♥➯♥ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû = a R ữợ tr M/N t❤➻ t❛ ❝â a ∈ annR (M/N )✳ ▼➔ M/N = (M/N )/(N/N ) ♥➯♥ ✈ỵ✐ ♠å✐ a ∈ R ữợ tr (M/N )/(N/N ) t a ∈ annR (M/N )/(N/N )✳ ❑❤✐ ✤â N/N ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M/N ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ ♠ỉ✤✉♥ (0) ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M/N ✳ ✹✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ✣à♥❤ ỵ R tr M Rổ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â✿ ✭✶✮ ▼ët ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❝õ❛ M ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì✳ ✭✷✮ ◆➳✉ N = N1 ∩ N2 ∩ · · · ∩ Nr ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü N ❝õ❛ M ✈➔ AssR M/Ni = {Pi } t❤➻ AssR M/N = {P1 , P2 , · · · , Pr }✳ ✭✸✮ ▼å✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü N ❝õ❛ M ✤➲✉ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ✳ (1) ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ N ❦❤æ♥❣ ♥❣✉②➯♥ tr♦♥❣ M t❤➻ N ❦❤↔ q✉② tr♦♥❣ M ✳ ❚❛ ❝â ♥❤➟♥ ①➨t✿ N ❦❤ỉ♥❣ ♥❣✉②➯♥ tr♦♥❣ M ⇔ ♠ỉ✤✉♥ (0) ❦❤ỉ♥❣ ❧➔ ♥❣✉②➯♥ tr♦♥❣ M/N ✭s✉② r❛ tø ❇ê ✤➲ ✸✳✶✹✮✳ ▼æ✤✉♥ (0) ❧➔ ❦❤↔ q✉② tr♦♥❣ M/N ⇔ N ❧➔ ❦❤↔ q✉② tr♦♥❣ M ✭s✉② r❛ tø ◆❤➟♥ ①➨t ✸✳✽✮✳ ❉♦ ✤â t❛ ❝â t❤➸ ❣✐↔ sû N = (0) tù❝ ❧➔ ❣✐↔ sû (0) ❦❤ỉ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ tr♦♥❣ M ✳ ❑❤✐ ✤â |AssR M | = 1✳ ▼➔ M = (0) ♥➯♥ t❛ s✉② r❛ |AssR M | 2✳ ●✐↔ sû 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 ✳ ❚❛ ❝â K1 , K2 = 0✳ ❍ì♥ ♥ú❛ ♥➳✉ K1 ∩ K2 = (0) t tỗ t = a + P1 R/P1 , = b + P2 ∈ R/P2 t❤ä❛ ♠➣♥ = f (a + P1 ) = g(b + P2 ) ∈ K1 ∩ K2 ✹✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❱➻ P1 = P2 ♥➯♥ ❝â t❤➸ ❝♦✐ P2 P❤↕♠ ◆❣å❝ ❉✐➺♣ P1 ✳ ●✐↔ sû α ∈ P1 \ P2 ✳ ❚❛ ❝â αf (a + P1 ) = αg(b + P2 ) ❚ø ✤â f (αa + P1 ) = g(αb + P2 ) = ❉♦ ✤â αb ∈ P2 ✭♠➙✉ t❤✉➝♥ ✈➻ α ∈ / P2 , b ∈ / P2 ✮✳ ❱➟② K1 ∩ K2 = (0)✳ ❑❤✐ ✤â (0) ❧➔ ❦❤↔ q✉② tr♦♥❣ M ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ N ❦❤↔ q✉② tr♦♥❣ M ✳ (2) ❚❛ ❝â t❤➸ ❣✐↔ sû N = (0) ✈➔ (0) = N = N1 ∩ N2 ∩ · · · ∩ Nr ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ ♠ỉ✤✉♥ ❝♦♥ t❤ü❝ sü N ❝õ❛ M ✳ ❚❛ ❝â ✤ì♥ ❝➜✉ M = M/(N1 ∩ N2 ∩ · · · ∩ Nr ) −→ M/N1 ⊕ M/N2 ⊕ · · · ⊕ M/Nr x + N1 ∩ N2 ∩ · · · ∩ Nr −→ (x + N1 , x + N2 + · · · , x + Nr ) ❙✉② r❛ AssR M ⊆ r AssR M/Ni = {P1 , P2 , · · · , Pr }✳ i=1 ❚❛ ❝❤ù♥❣ ♠✐♥❤ P1 ∈ AssR M ✳ ❱➻ N ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ♥➯♥ N2 ∩ · · ã Nr = N = (0) õ tỗ t↕✐ = x ∈ N2 ∩ · · · ∩ Nr ✳ ❚❛ ❝â annR x = :R x = N1 :R x✳ ▼➔ N1 ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ♥➯♥ t❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✸✳✹ t❛ ❝â P1 = annR (M/N1 )✳ ❚ø ✤â t❤❡♦ ▼➺♥❤ t tỗ t v tọ P1 v M ⊆ N1 ✳ ❉♦ ✤â P1v ⊆ N1 :R x = :R x ⇒ P1 v x = 0✳ ❚ø ✤â ✈➔ tø x = 0✱ ∃i s❛♦ ❝❤♦ P1i x = ♥❤÷♥❣ P1i+1 x = 0✳ ❈❤å♥ = y ∈ P1i x ∈ N2 ∩ · · · ∩ Nr ✳ ❑❤✐ ✤â P1 y = ♥➯♥ t❛ s✉② r❛ P1 ⊆ annR y ✳ ❍ì♥ ♥ú❛ ✈ỵ✐ a ∈ annR y t❛ ❝â ay = ∈ N1 ✳ ▼➔ y ∈ / N1 ✱ N1 ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✸✳✷ t❤➻ a ∈ ✹✺ annR (M/N1 ) = P1 ✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ❉♦ ✤â P1 = annR y ✳ ❱➟② P1 ∈ AssR M ✳ ❚÷ì♥❣ tü ✈ỵ✐ ❝→❝ Pi ❝á♥ ❧↕✐ t❛ s✉② r❛ AssR M = {P1 , P2 , · · · , Pr }✳ (3)✳ ❚ø ▼➺♥❤ ✤➲ ✸✳✾ ✈➔ tø (1) t❛ s✉② r❛ ♠å✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ M ✤➲✉ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ✹✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P❤↕♠ ◆❣å❝ ❉✐➺♣ ❑➳t ❧✉➟♥ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ✧■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✧ ✳ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t ❝õ❛ ♠➻♥❤ ♠ët ❝→❝❤ ❤➺ t❤è♥❣✱ rã r➔♥❣ ✈➲ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ♠ët ♠æ✤✉♥ ✈➔ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ❑❤â❛ ❧✉➟♥ ✤➣ ✤↕t ✤÷đ❝ ♠ư❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ư ✤➲ r❛✳ ❚✉② ♥❤✐➯♥ ❞♦ t❤í✐ ❣✐❛♥ ❝á♥ ❤↕♥ ❝❤➳ ✈➔ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ rt ữủ ỳ ỵ õ õ qỵ t ổ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ✤➛② ✤õ ✈➔ t ỡ rữợ t tú õ ♠ët ❧➛♥ ♥ú❛ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❣✐→♦ ✣é ❱➠♥ ❑✐➯♥ ✤➣ t t ữợ ú ù t ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❍♦➔♥❣ ❳✉➙♥ ❙➼♥❤✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❬✷❪ ❉÷ì♥❣ ◗✉è❝ ❱✐➺t✱ ❈ì s ỵ tt ữ ❙❤✐r♦ ●♦t♦✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❤♦♠♦❧♦❣✐❝❛❧ ♠❡t❤♦❞s ✐♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ✳ ❬✹❪ ❍✐❞❡②✉❦✐ ▼❛ts✉♠✉r❛✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡s ◆❛❣♦②❛ ❯♥✐✈❡rs✐t② ✳ ✹✽ ... ♠ỉ✤✉♥ t❤÷ì♥❣✱ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ❝õ❛ ♠ỉ✤✉♥✱ ❞➣② ❦❤ỵ♣✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ♣❤ư❝ ✈ư ự ỵ ữỡ s ữỡ tố ❧✐➯♥ ❦➳t ❈❤÷ì♥❣ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ✈➲ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝ò♥❣ ❝→❝ ♠➺♥❤ ✤➲✱ ✤à♥❤ ❧➼ ❧✐➯♥... ✧✳ ❑❤✐ ✤â X ❝â ♣❤➛♥ tû ❝ü❝ ✤↕✐✳ ◆❤➟♥ ①➨t ✶✳✷✹✳ ▼å✐ ✈➔♥❤ ❝â ✤ì♥ ✈à ✤➲✉ ❝â ✐✤➯❛♥ ❝ü❝ õ ổ tỗ t tố ✣à♥❤ ♥❣❤➽❛ ✶✳✷✺✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â √ I = {x ∈ R | ∃n ∈ N∗ : xn ∈ I} ❧➔ ♠ët... A✳ ▼➺♥❤ ✤➲ ✶✳✸✺✳ ◆➳✉ f : X −→ Y ✱ g : Y Z ỗ t t ①↕ ❤ñ♣ t❤➔♥❤✮ g ◦ f : X −→ Z ụ ởt ỗ õ tốt P ữợ ✤➙② ❧➔ ♠ët tr♦♥❣ ❝→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ỗ f : X Y ởt ỗ õ ❍↕t ♥❤➙♥