Luận văn thạc sĩ định lý phân tích nguyên sơ noether và ý nghĩa hình học

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ●■⑩P ❚❍➚ ❚❍Õ❨ ✣➚◆❍ ▲Þ P❍❹◆ ❚➑❈❍ ◆●❯❨➊◆ ❙❒ ◆❖❊❚❍❊❘ ❱⑨ Þ ◆●❍➒❆ ❍➐◆❍ ❍➴❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ì P P ế ị P ◆●❯❨➊◆ ❙❒ ◆❖❊❚❍❊❘ ❱⑨ Þ ◆●❍➒❆ ❍➐◆❍ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ số ỵ tt số số ữớ ữợ ❚❙✳ P❍❸▼ ❍Ị◆● ◗❯Þ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ❝❤➾ ró ỗ ố t ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ●✐→♣ ❚❤à ❚❤õ② ✐ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ sữ rữợ ❦❤✐ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ tæ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤✱ s➙✉ s➢❝ tỵ✐ P ũ ỵ t ữớ trỹ t ữợ ❞➝♥✱ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❜❛♥ ỏ s qỵ t ổ tr ❦❤♦❛ ❚♦→♥✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❝❛♦ ❤å❝ ❚♦→♥ ❦✷✶❜ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐✱ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣✳ ◗✉❛ ✤➙②✱ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ s s tợ ữớ t tr ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ▼➦❝ ❞ò ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ✈➝♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât ✈➔ ❤↕♥ ❝❤➳✳ ❚æ✐ r➜t ♠♦♥❣ ♥❤➟♥ ữủ ỳ ỵ õ õ qỵ t ❝ỉ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ●✐→♣ ❚❤à ❚❤õ② ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲í✐ ❝↔♠ ì♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✷✳ ❱➔♥❤ ✈➔ ♠ỉ✤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳ ❱➔♥❤ ✈➔ ♠æ✤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ìợ ❝õ❛ tr♦♥❣ ✈➔♥❤ ◆♦❡t❤❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ tr ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✶✳ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✷✳ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹✳ ■✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✹✳✶✳ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t ❤ú✉ ❤↕♥ ✈➔ ✐✤➯❛♥ ❝↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✐✐✐ ▼Ð ✣❺❯ ▼ët tr ỳ ỵ t ỵ ỡ số ỵ ❦❤➥♥❣ ✤à♥❤ r➡♥❣✿ ▼å✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✤➲✉ ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ t➼❝❤ ❝→❝ ❧ơ② t❤ø❛ ❝õ❛ ❝→❝ sè ♥❣✉②➯♥ tố ỵ t sỡ tr sỹ rở ỵ ỡ số ởt ợ rở ợ tr ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❊♠♠② ◆♦❡t❤❡r ✈➔♦ ✤➛✉ t❤➳ ❦✛ ❳❳ ✈➔ ✤➣ trð t❤➔♥❤ ♥➲♥ t↔♥❣ ❝❤♦ ✣↕✐ sè ❣✐❛♦ ❤♦→♥ ✈➔ ❤➻♥❤ ❤å❝ ✤↕✐ sè✳ ❈❤♦ ♠ët ✈➔♥❤ tr ỵ r t➼❝❤ ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ ✐✤➯❛♥ sỡ ữỡ ự ỵ t➼❝❤ ♥❣✉②➯♥ ◆♦❡t❤❡r ❧➔✿ ▼å✐ t➟♣ ✤↕✐ sè ✤➲✉ ❧➔ ❤ñ♣ ❝õ❛ ❤ú✉ ❤↕♥ t➟♣ ✤↕✐ sè ❜➜t ❦❤↔ q ỵ q trồ ỵ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ◆♦❡t❤❡r t→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ✤➦t t t õ ỵ ❝õ❛ ❝→❝ ✤è✐ t÷đ♥❣ ❧✐➯♥ q✉❛♥✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ✈✐➳t t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ 1✿ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ✣↕✐ sè ❣✐❛♦ ❤♦→♥ ♥❤÷✿ ❱➔♥❤✱ ♠ỉ✤✉♥✱ ✐✤➯❛♥ ♥❣✉②➯♥ tè✱ ✤à❛ ♣❤÷ì♥❣ ❤â❛✱ ❜ê ✤➲ ◆❛❦❛②❛♠❛✳ ❈❤÷ì♥❣ 2✿ ❚r➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✈➲ ✈➔♥❤✱ ♠ỉ✤✉♥ ◆♦❡t❤❡r ỵ ỡ s rt ú tổ tr ỵ t sỡ tr t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ✈➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝✱ ✐✤➯❛♥ ❝↕♥❤ ✤÷đ❝ ✤÷❛ r❛ ð ❝✉è✐ ❝❤÷ì♥❣ ❞ị♥❣ ✤➸ ♠✐♥❤ ỵ t sỡ ữỡ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ①➨t ✈➔♥❤ ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✶✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✱ ♠ët t➟♣ ❝♦♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R ♥➳✉ t❤ä❛ ♠➣♥✿ ✭✐✮ I ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ R ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ +❀ ✭✐✐✮ ❱ỵ✐ ♠å✐ ♣❤➛♥ tû x t❤✉ë❝ R✱ ♠å✐ ♣❤➛♥ tû a t❤✉ë❝ I t❤➻ xa ∈ I ✭ax ∈ I ✮✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✷✳ ❈❤♦ p ❧➔ ♠ët ✐✤➯❛♥ t❤➟t sü ❝õ❛ R✳ ❑❤✐ ✤â p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➳✉ ✈ỵ✐ ♠å✐ x, y t❤✉ë❝ R t❤♦↔♥ ♠➣♥ xy ∈ p t❤➻ x ∈ p ❤♦➦❝ y ∈ p✳ ❚❛ ❦➼ ❤✐➺✉ ❙♣❡❝✭❘✮ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ❱➼ ❞ö ✶✳✵✳✸✳ ❚❛ ❝â Spec(Z) = {(0), pZ| p ❧➔ ♠ët sè ♥❣✉②➯♥ tè}✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✹✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ õ R/I ợ ữủ ữ s❛✉✿ (x + I) (y + I) = xy + I; ∀x, y ∈ R ❧➔ ♠ët ✈➔♥❤✳ ❱➔♥❤ R/I ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ R t❤❡♦ ✐✤➯❛♥ I ✳ ✷ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✺✳ ▼ët ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ♥➳✉ R = ✈➔ ♥➳✉ x, y = t❤➻ xy = 0✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ♠✐➲♥ ♥❣✉②➯♥ t❛ t❤➜② ♥❣❛② r➡♥❣ ✐✤➯❛♥ (0) ❝õ❛ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tố qt t õ s ỵ ■✤➯❛♥ p ❝õ❛ ♠ët ✈➔♥❤ R ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈➔♥❤ t❤÷ì♥❣ R/p ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✼✳ ▼ët ✐✤➯❛♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ ♥➳✉ I = R ✈➔ ♥â ❦❤æ♥❣ ❝❤ù❛ tr♦♥❣ ❜➜t ❦ý ♠ët ✐✤➯❛♥ t❤ü❝ sỹ ỵ Max(R) t ủ tt ❝↔ ❝→❝ ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R ◆❤➟♥ ①➨t ✶✳✵✳✽✳ ✭✐✮ ▼å✐ ✐✤➯❛♥ tè✐ ✤↕✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ✭✐✐✮ ◆➳✉ R ❧➔ ♠ët tr÷í♥❣ t❤➻ ✐✤➯❛♥ (0) ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐✳ ✭✐✐✐✮ ■✤➯❛♥ I ❧➔ tè✐ ✤↕✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R/I ❧➔ ♠ët tr÷í♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✾✳ ▼ët ✈➔♥❤ R ❝â ❞✉② ♥❤➜t ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ m ữủ ữỡ ỵ (R, m) ữợ ởt số t q trồ ❝õ❛ ✐✤➯❛♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✶✵✳ ❈❤♦ I ✈➔ J ❧➔ ❤❛✐ ✐✤➯❛♥✳ ❑❤✐ ✤â t❛ ✤à♥❤ ♥❣❤➽❛✿ ✭✐✮ P❤➨♣ ❝ë♥❣ ❝→❝ ✐✤➯❛♥✱ I + J = {a + b |a ∈ I, b ∈ J }✳ ✭✐✐✮ P❤➨♣ ❣✐❛♦ ❝→❝ ✐✤➯❛♥✱ I ∩ J = {a|a ∈ I ✈➔ a ∈ J}✳ ✭✐✐✐✮ P❤➨♣ ❝❤✐❛ ✐✤➯❛♥✱ I : J = {x| xJ ⊆ I} ⊇ I ✳ √ ✭✐✈✮ P❤➨♣ ❧➜② ❝➠♥ ✐✤➯❛♥✱ I = {x| ∃n : xn ∈ I}✳ ◆❤➟♥ ①➨t ✶✳✵✳✶✶✳ ◆➳✉ J = (a1, , ak ) t❤➻ I : J = i=1 (I : ai)✳ k ❱➼ ❞ö ✶✳✵✳✶✷✳ ❳➨t R = Z✱ I ✈➔ J ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ Z✱ I = (a)✱ J = (b)✳ ❑❤✐ ✤â✿ ✸ I + J = {ax + by |x, y ∈ Z } = ×❈▲◆ (a, b) Z; I ∩ J = ❇❈◆◆ (a, b) Z❀ ✳ ✳ I : J = x xb✳✳a = x✳✳ ×❈▲◆a (a,b) = ×❈▲◆a (a,b) ✳ √ √ I = {p |∃n : pn ∈ I } ✈ỵ✐ p = pα1 pαk k t❤➻ I = p1 pk Z✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✶✸✳ ❈➠♥ ❝õ❛ ❧➔ t➟♣ ❤ñ♣ ❝→❝ ♣❤➛♥ tû ❧ơ② ❧✐♥❤ ❝õ❛ R ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ Nil(R) ❚❛ ❝â ♠è✐ ❧✐➯♥ ❤➺ ❝õ❛ Nil(R) ✈ỵ✐ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥❤÷ s❛✉✿ √ ❚r♦♥❣ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ R t õ Nil(R) = = p ỵ ✶✳✵✳✶✹✳ p∈SpecR ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✶✺✳ ■✤➯❛♥ q ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ♥➳✉ q = R ✈➔ ✈ỵ✐ ♠å✐ x.y t❤✉ë❝ q ✈➔ y ❦❤ỉ♥❣ t❤✉ë❝ q t xn tở q ợ ởt số ữỡ n ♥➔♦ ✤â✳ ❱➼ ❞ö ✶✳✵✳✶✻✳ ❚r♦♥❣ t➟♣ sè ♥❣✉②➯♥ Z✱ ✈ỵ✐ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè t❤➻ pα Z ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ❝õ❛ Z ▼➺♥❤ ✤➲ ✶✳✵✳✶✼✳ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ❝õ❛ R ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (0) ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ❝õ❛ R/q✳ ✭✐✐✮ ✭✐✐✐✮ ✭✐✮ q sỡ R t ữợ ❝õ❛ ✤➲✉ ❧➔ ❧ô② ❧✐♥❤✳ ◆➳✉ q ❧➔ ✐✤➯❛♥ sỡ t q tố ỵ ỵ tr tố s ❧➔ ✤ó♥❣ ❝❤♦ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ R✳ ✭✐✮ ❈❤♦ p1, p2, , pn ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ✈➔ a ❧➔ n♠ët ✐✤➯❛♥ ❝õ❛ R✳ ●✐↔ sû a ⊂ pi ✈ỵ✐ ♠å✐ i = 1, 2, , n ❦❤✐ ✤â a ⊂ pi ✳ i=1 ✭✐✐✮ ❈❤♦ an1, a2, , an ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ✈➔ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ◆➳✉ ⊆p t❤➻ ❦❤✐ ✤â tỗ t ởt số i s p✳ ❍ì♥ i=1 ♥ú❛✱ ❦❤✐ n =p i=1 t❤➻ tỗ t số i s = p ✹ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ n✳ ❑❤✐ n = t❤➻ ❦➳t ❧✉➟♥ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ●✐↔ sû (i) ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ tr÷í♥❣ ❤đ♣ n − 1✱ tù❝ a ⊂ ♥❤ú♥❣ ♣❤➛♥ tû xt ∈ a\ i=t pi ✱ i=t pi ✱ n ✈ỵ✐ ♠å✐ t = 1, 2, , n tỗ t ợ t = 1, , n✳ ◆➳✉ xt ∈ / pt ✈ỵ✐ ♠ët pi ✈➔ ♠➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚r→✐ ❧↕✐✱ sè t ♥➔♦ ✤â✱ s✉② r❛ xt ∈ a\ i=1 ❣✐↔ sû xt ∈ pt ✱ ✈ỵ✐ ♠å✐ t = 1, 2, , n✳ ❳➨t ♣❤➛♥ tû n x= x1 x2 xi xn i=1 n tr♦♥❣ ✤â x = x1 x2 xi xn ❦➼ ❤✐➺✉ ❝❤♦ t➼❝❤ ❝→❝ ♣❤➛♥ tû x1 , , xn s❛✉ i=1 ❦❤✐ ❜ä ✤✐ ♣❤➛♥ tû xi ✳ ❘ã r➔♥❣ x ∈ a✳ ❚r♦♥❣ ❦❤✐✱ ♥➳✉ x ∈ pi ✱ ❦➨♦ t❤❡♦ x1 x2 xi xn ∈ pi ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❝→❝❤ ❝❤å♥ xt ✈➔ ❞♦ ✤â (i) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✭✐✐✮ ●✐↔ sû ♠➺♥❤ ✤➲ s❛✐✱ tù❝ ❧➔ ⊂ p✱ ✈ỵ✐ ♠å✐ i = 1, 2, , n õ tỗ t ỳ ♣❤➛♥ tû yi ∈ \p✱ ✈ỵ✐ ♠å✐ i = 1, 2, , n✳ ✣➦t y = y1 yn ✱ t❛ s✉② r❛ y = n ⊆ p✳ ❱➟② tỗ t ởt số i s yi ∈ p ✈➔ i=1 ✤✐➲✉ ♥➔② tr→✐ ✈ỵ✐ ❝→❝❤ ❝❤å♥ yi ✳ ❇➙② ❣✐í✱ ♥➳✉ n = p t❤➻ ✈ỵ✐ ❝❤➾ sè i ð i=1 tr➯♥ t❛ ❝â p ⊆ ⊆ p✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä = p ỵ ữủ ự ✭❇ê ✤➲ ❩♦r♥✮✳ ❈❤♦ A ❧➔ ♠ët t➟♣ ❦❤→❝ ré♥❣ ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü ≤✳ ●✐↔ sû ♠å✐ ❞➣② t➠♥❣ ❝→❝ ♣❤➛♥ tû tr♦♥❣ A a1 ≤ a2 ≤ ≤ an ≤ ≤ a ✤➲✉ ❝â ♣❤➛♥ tỷ tr tự tỗ t a A s❛♦ ❝❤♦ ≤ a ✈ỵ✐ ♠å✐ i ≥ t tr A tỗ t tỷ tố q✉↔ ✶✳✵✳✷✵✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✳ ❑❤✐ ✤â R ❧✉æ♥ ❝â ✐✤➯❛♥ tè✐ ✤↕✐✳ ✺ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✸✳ ◆➳✉ Ass(M/N ) = {p} t❛ ♥â✐ r➡♥❣ N ⊂ M ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ p✲♥❣✉②➯♥ ❝õ❛ M ỵ N N ởt ♠ỉ✤✉♥ ❝♦♥ p✲♥❣✉②➯♥ ❝õ❛ M t❤➻ ❦❤✐ ✤â N ∩ N ❝ơ♥❣ ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ p✲♥❣✉②➯♥ ❝õ❛ M ự t ỗ : M/(N ∩ N ) → M/N ⊕ M/N x + (N ∩ N ) → (x + N ) ⊕ (x + N ) ❧➔ ♠ët ✤ì♥ ❝➜✉✳ ❱➟② Ass M/(N ∩ N ) ⊂ Ass M/N ∪ Ass M/N ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✺✳ ✭✐✮ ▼ët ✐✤➯❛♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ = {p} ❜➜t ❦❤↔ q✉② ♥➳✉ I ❦❤æ♥❣ ❧➔ ❣✐❛♦ ❝õ❛ ❤❛✐ ✐✤➯❛♥ ❝❤ù❛ ♥â t❤ü❝ sü✳ ✭✐✐✮ ▼ët ♠ỉ✤✉♥ ❝♦♥ N ❝õ❛ M ✤÷đ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ♥➳✉ ♥â ❦❤æ♥❣ ❧➔ ❣✐❛♦ ❝õ❛ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❝❤ù❛ ♥â t❤ü❝ sü✳ ◆❤➟♥ ①➨t ✷✳✸✳✻✳ ▼å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè ✤➲✉ ❧➔ ❜➜t ❦❤↔ q✉② ✈➔ tr♦♥❣ ✈➔♥❤ ◆♦❡t❤❡r t❤➻ ♠å✐ ✐✤➯❛♥ ❜➜t ❦❤↔ q✉② ❧➔ ♥❣✉②➯♥ sì✳ ▼➺♥❤ ✤➲ ✷✳✸✳✼✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ♠å✐ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② N ❝õ❛ M ✤➲✉ ❧➔ ♥❣✉②➯♥ sì✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❛② M ❜ð✐ M/N t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ N = 0✳ ỵ Ass(M ) õ t t ♣❤➛♥ tû p1 , p2 ✳ ❑❤✐ ✤â M ❝❤ù❛ ♠ỉ✤✉♥ ❝♦♥ Ki ✤➥♥❣ ❝➜✉ tỵ✐ A/pi ✈ỵ✐ i = 1, 2✳ ❚ø Ann(x) = pi ✈ỵ✐ x ❦❤ỉ♥❣ ❧➔ ÷ỵ❝ ❝õ❛ ✈➔ x ∈ Ki ✳ ❚❛ ❝â K1 ∩ K2 = 0✱ ✈➔ ❞♦ ✤â ❧➔ ❦❤↔ q✉②✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ❜❛♥ ỵ R ởt ◆♦❡t❤❡r ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ✤➲✉ ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ N ❝õ❛ M ❦❤æ♥❣ ❧➔ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝õ❛ ♥❤ú♥❣ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ✈➔ ❣✐↔ sû ❉♦ M ❧➔ ởt Rổ tr tỗ t tr = ởt ♣❤➛♥ tû ❝ü❝ ✤↕✐ N ✳ ❑❤✐ ✤â N ♣❤↔✐ ổ q tự tỗ t ổ ❝♦♥ N1 ⊃ N, N2 ⊃ N s❛♦ ❝❤♦ N = N1 ∩ N2 ✳ ❱➻ N1 , N2 ∈ / ♥➯♥ ❝❤ó♥❣ ❧➔ ❣✐❛♦ ❤ú✉ ❤↕♥ ♥❤ú♥❣ ♠ỉ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❞♦ ✤â N ❝ơ♥❣ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉②✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t N ∈ ỵ ữủ ự t tr ỵ t sỡ tr ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✾✳ ▼ët ♠ỉ✤✉♥ ❝♦♥ N ❝õ❛ M ✤÷đ❝ ❣å✐ ❧➔ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ♥➳✉ ♥â ✈✐➳t ✤÷đ❝ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì✱ tù❝ ❧➔ N = N1 ∩ N2 ∩ · · · ∩ Nr ✈ỵ✐ Ni ❧➔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì✳ ▼ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ♥➳✉ t❛ ❦❤ỉ♥❣ t❤➸ ❜ä ❜➜t ❦ý ♠ët ♠ỉ✤✉♥ ♥❣✉②➯♥ ♥➔♦ tr♦♥❣ ♣❤➙♥ t➼❝❤ ✤â✱ tù❝ ❧➔ N = N1 ∩ · · · ∩ Ni−1 ∩ Ni+1 · · · ∩ Nr ✈ỵ✐ ♠å✐ i = 1, , r ỵ sỷ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥✳ ▼å✐ ♠æ✤✉♥ ❝♦♥ N ❝õ❛ M ✤➲✉ ❝â ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥✳ ❍ì♥ ♥ú❛✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ ①✉➜t ❤✐➺♥ tr♦♥❣ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ✤➲✉ ù♥❣ ✈ỵ✐ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♣❤➙♥ ❜✐➺t✳ ✭✐✐✮ ◆➳✉ N = N1 ∩ N2 ∩ ∩ Nr ✈ỵ✐ Ass(M/Ni ) = {pi } ❧➔ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ❝õ❛ N t❤➻ Ass(M/N ) = {p1, pr }✳ ✭✐✐✐✮ ◆➳✉ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t tè✐ t✐➸✉ ❝õ❛ M/N t❤➻ t❤➔♥❤ ♣❤➛♥ p✲♥❣✉②➯♥ ①✉➜t ❤✐➺♥ tr♦♥❣ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ❧➔ ϕ−1 p (Np )✱ ð ✤â ϕp : M Mp ỗ t õ t sỡ N tữỡ ự ợ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ♥❤ä ♥❤➜t ❧➔ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳ ✭✐✮ ✷✸ ❈❤ù♥❣ ♠✐♥❤✳ (i) ❚❤❡♦ ✣à♥❤ ỵ t õ N ữủ t t ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ♥➯♥ ♥â ❝ơ♥❣ ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❞♦ ▼➺♥❤ ✤➲ ✷✳✸✳✼✳ ❚❛ ❝â t❤➸ ❜ä ✤✐ ♥❤ú♥❣ ♠æ✤✉♥ ❝♦♥ ❦❤ỉ♥❣ ❝➛♥ t❤✐➳t ❝õ❛ ♣❤➙♥ t➼❝❤ ♥➔② ✤➸ t❤✉ ✤÷đ❝ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥✳ ◆➳✉ Ni ✈➔ Nj ✤➲✉ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ p✲♥❣✉②➯♥ t❤➻ Ni ∩ Nj ❝ơ♥❣ ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ p✲♥❣✉②➯♥ t❤❡♦ ỵ ữ t õ t õ ỳ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ ù♥❣ ✈ỵ✐ ❝ị♥❣ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧➔ ♠ët t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ ❝❤✉♥❣✳ ❑❤➥♥❣ ✤à♥❤ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ (ii) ❚❤❛② M ❜➡♥❣ M/N t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ N = 0✳ ❉♦ = N1 ∩ ∩Nr t❛ ❝â M ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M/N1 ⊕ ⊕ M/Nr ✈➟② Ass(M ) ⊂ Ass r ⊕ M/Ni r Ass M/Ni = {p1 , , pr } = i=1 i=1 ❱➻ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❧➔ rót ❣å♥ ♥➯♥ N2 ∩ ∩ Nr = ♥➯♥ t❛ ❝â ♠ët ♣❤➛♥ tû = x ∈ N2 ∩ ∩ Nr ✳ ❚❛ ❝â Ann(x) = : x = N1 : x✳ ◆❤÷♥❣ AnnM/N1 = N1 : M ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ sỡ p1 t ỵ 2.3 pv1 M ⊂ N1 ✈ỵ✐ ♠ët sè v > 0✳ ◆➯♥ pv1 x = õ tỗ t i i s❛♦ ❝❤♦ pi1 x = ♥❤÷♥❣ pi+1 x = 0✳ ❈❤å♥ ♠ët ♣❤➛♥ tû = y ∈ p1 x t❛ ❝â p1 y = 0✳ ❚✉② ♥❤✐➯♥✱ tø y ∈ N2 ∩ ∩ Nr t❤ä❛ ♠➣♥ r➡♥❣ y ∈ / N1 ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ t❛ ❝â Ann(y) ⊂ p1 ♥➯♥ p1 = Ann(y) ✈➔ p1 ∈ Ass(M )✳ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ tr➯♥ ❝❤♦ ❝→❝ pi ❝á♥ ❧↕✐ t❛ ❝â pi ⊂ Ass(M )✱ ✈ỵ✐ i = 1, , r✳ ✭✐✐✐✮ ●✐↔ sû r➡♥❣ N = N2 ∩ ∩ Nr ❧➔ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ♥❤÷ ✭✐✮ ✈➔ N1 ❧➔ t❤➔♥❤ ♣❤➛♥ p✲♥❣✉②➯♥ ✈ỵ✐ p = p1 ✳ ❚❛ ❝â Np = (N1 )p ∩ ∩ (Nr )p ✱ ✈➔ ✈ỵ✐ ♠å✐ i > ♠ët ❧ô② t❤ø❛ ❝õ❛ pi s➩ ❜à ❝❤ù❛ tr♦♥❣ Ann(M/Ni )✳ ▼➔ pi p1 ♥➯♥ (M/Ni )p = 0✳ ❉➝♥ ✤➳♥ (Ni )p = Mp ♥➯♥ ϕ−1 Np = (N1 )p ✳ ❱➟② ϕ−1 (N1 )p = N1 ỵ ữủ ự p (Np ) = p ỵ tr ởt ◆♦❡t❤❡r R ✤➲✉ ❝â ♠ët ♣❤➙♥ ✷✹ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥✳ ❚ù❝ ❧➔ ✈ỵ✐ ♠å✐ ✐✤➯❛♥ I ❝õ❛ R t❛ ❝â I = q1 ∩ · · · ∩ qr Ð ✤➙②✱ qi ❧➔ pi✲♥❣✉②➯♥ ✈➔ Ass(R/I) = {p1, , pr } ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✷✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ✭✐✮ ❚❛ ❣å✐ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ♥❤ä ♥❤➜t ❝õ❛ M/N ❧➔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t tè✐ t✐➸✉ ❝õ❛ N tr♦♥❣ M ✳ ❈→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝á♥ ❧↕✐ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥❤ó♥❣✳ ✭✐✐✮ ❈→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ ①✉➜t ❤✐➺♥ tr♦♥❣ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N tr♦♥❣ M ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ tè✐ t✐➸✉ ✈➔ t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ ❝õ❛ N ✱ t÷ì♥❣ ù♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ N = t❤➻ t❛ ♥â✐ ✤â ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ tè✐ t✐➸✉✱ t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ ❝õ❛ M ỵ t sỡ t t❤➜② ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ tè✐ t✐➸✉ ❧➔ ❞✉② ♥❤➜t✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❣✉②➯♥ ♥❤ó♥❣ ❧➔ ❦❤ỉ♥❣ ❞✉② ♥❤➜t✳ ❱➼ ❞ư ✷✳✸✳✶✸✳ ❳➨t K ❧➔ ♠ët tr÷í♥❣ ✈➔ R = K[x, y] ❧➔ ♠ët ✈➔♥❤ ✤❛ t❤ù❝✳ ❑❤✐ ✤â✱ ✐✤➯❛♥ I = (x2 , xy) ❝â ❝→❝ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ s❛✉✿ (x2 , xy) = (x) ∩ (x2 , y) = (x) ∩ (x2 , xy, y ) ❚❛ ❝â Ass(R/I) = {(x), (x, y)} ✈➔ (x) ❧➔ t❤➔♥❤ ♣❤➛♥ tè✐ t✐➸✉ ❝á♥ (x2 , y) ✈➔ (x2 , xy, y ) ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ ❝õ❛ I ✳ ✷✳✸✳✷✳ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ①➨t K ❧➔ ♠ët tr÷í♥❣ ✈➔ R = K[x1 , , xn ] = K[x]✳ ❱ỵ✐ ♠å✐ ❤å ✤❛ t❤ù❝ {fi }i∈A t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❤å ✤❛ t❤ù❝ ♥➔② ❧➔ Z({fi }i∈A ) = {x ∈ K n |fi (x) = 0, ∀i ∈ A} ✷✺ ✣➦t I ❧➔ ✐✤➯❛♥ s✐♥❤ ❜ð✐ ❤å ✤❛ t❤ù❝ tr➯♥✱ t❛ t❤➜② t➟♣ ♥❣❤✐➺♠ tr➯♥ ❝ô♥❣ ❝❤➼♥❤ ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ✐✤➯❛♥ Z(I) ♥➯♥ tø ♥❛② t❛ ❝❤➾ ①➨t ✤➳♥ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ✐✤➯❛♥✳ ▼ët sè t➼♥❤ ❝❤➜t s❛✉ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✹✳ ✭✐✮ ◆➳✉ I ⊆ J t❤➻ Z(J) ⊆ Z(I)✳ √ ✭✐✐✮ Z(I) = Z( I)✳ ✭✐✐✐✮ Z(I) ∪ Z(J) = Z(I ∩ J)✳ ✭✐✈✮ ∩Z(Ii ) = Z( Ii ) ✈ỵ✐ ♠å✐ ❤å ✐✤➯❛♥ {Ii}✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✺✳ ❚➟♣ ♥❣❤✐➺♠ Z(I) ❝õ❛ ♠ët ✐✤➯❛♥ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ✤↕✐ sè✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✸✳✶✹ t❛ t❤➜②✱ ❤ñ♣ ❤❛✐ t➟♣ ✤↕✐ sè ✈➔ ❣✐❛♦ ❝õ❛ ♠ët ❤å ❝→❝ t➟♣ ✤↕✐ sè ❝ô♥❣ ❧➔ t➟♣ ✤↕✐ sè ♥➯♥ t❛ ❝â t❤➸ tr❛♥❣ ❜à ♠ët tæ ♣æ ❝❤♦ ❦❤æ♥❣ ❣✐❛♥ K n ❜➡♥❣ ❝→❝❤ ❝♦✐ ❝→❝ t➟♣ ✤↕✐ sè ❧➔ ❝→❝ t➟♣ ✤â♥❣✳ ❚æ ♣æ ♥➔② tæ ♣æ ❩❛r✐s❦✐✳ ❑❤æ♥❣ ❣✐❛♥ K n ợ tổ ổ rs ữủ ổ n ỵ An ữủ t V ởt t ữủ tũ ỵ tr An ỵ IV = {f K[x]|f (a) = 0, ∀a ∈ V } ❈â t❤➸ t❤➜② ♥❣❛② IV ❧➔ ✐✤➯❛♥ ✈➔ ❧➔ ✐ ✤➯❛♥ ❧ỵ♥ ♥❤➜t ❝â t➟♣ ♥❣❤✐➺♠ ❝❤ù❛ V ✳ ❚❛ ❣å✐ IV ❧➔ ✐✤➯❛♥ ❝õ❛ t➟♣ ✤✐➸♠ V tr♦♥❣ K[x]✳ ●å✐ V ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ✤↕✐ sè ❝❤ù❛ V ✱ ❦❤✐ ✤â V ❝ô♥❣ ❧➔ ♠ët t➟♣ ✤↕✐ sè ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ V ✳ ❚❛ ❝â ❜ê ✤➲ s❛✉ ❇ê ✤➲ ✷✳✸✳✶✻✳ ❈❤♦ V ❧➔ ởt t tũ ỵ tr An õ ởt ✐✤➯❛♥ ❝➠♥✳ ✭✐✐✮ V = Z(IV )✳ ✭✐✐✐✮ IV = IV ✳ ✭✐✮ IV ✷✻ ❳➨t V ❧➔ ♠ët t➟♣ ✤↕✐ sè✱ t❛ ❝â V = V ♥➯♥ V = Z(IV )✳ ◆❤÷ ✈➟② t❛ ❝â t÷ì♥❣ ù♥❣ − ❣✐ú❛ ❝→❝ t➟♣ ✤↕✐ sè ✈➔ ❝→❝ ✐✤➯❛♥ ❝➠♥ ❞↕♥❣ IV ✳ ❚❛ t❤÷í♥❣ t➻♠ ❝→❝❤ ♣❤➙♥ t➼❝❤ ♠ët t➟♣ ✤↕✐ sè t❤➔♥❤ ❤ñ♣ ❝→❝ t➟♣ ✤↕✐ sè ♥❤ä ❤ì♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➟♣ ✤↕✐ sè ♥➔②✳ ◆➳✉ t❛ ❦❤æ♥❣ t❤➸ ♣❤➙♥ t➼❝❤ ♠ët t➟♣ ✤↕✐ sè ❦❤→❝ ré♥❣ t❤➔♥❤ ❤đ♣ ❤❛✐ t➟♣ ✤↕✐ sè ♥❤ä ❤ì♥ t❤➻ t❛ ❣å✐ t➟♣ ✤↕✐ sè ✤â ❧➔ t➟♣ ❜➜t ❦❤↔ q số tữỡ ự ợ t t q tố ỵ ❚➟♣ ✤↕✐ sè V ❧➔ ❜➜t ❦❤↔ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ IV ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ◗✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ t➟♣ ✤↕✐ sè ✈➔ ❝→❝ ✐✤➯❛♥ ❝➠♥ ð tr➯♥ ởt s ỵ ổ t s rt ỵ K trữớ õ số ợ I K[x] √ t❛ ❝â IZ(I) = I✳ ❚❤æ♥❣ q✉❛ ✣à♥❤ ỵ ổ rt t õ q K ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè✳ ❚❛ ❝â ♠ët s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ t➟♣ ✤↕✐ sè ✈➔ ❝→❝ ✐✤➯❛♥ ❝➠♥✱ ✈➔ s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ t➟♣ ✤↕✐ sè ❜➜t ❦❤↔ q✉② ✈➔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❱➻ K[x1 , , xn ] ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ♥➯♥ ♠å✐ ✐✤➯❛♥ ❝õ❛ ♥â ✤➲✉ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ I ❧➔ ♠ët ✐✤➯❛♥ ❝➠♥ t❤➻ ♥â ✤÷đ❝ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❤ú✉ ❤↕♥ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ữỡ ự ỵ t sỡ ữ s ỵ r ổ An ♠å✐ t➟♣ ✤↕✐ sè ✤➲✉ ♣❤➙♥ t➼❝❤ ✤÷đ❝ t❤➔♥❤ ❤ñ♣ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ✤↕✐ sè ❜➜t ❦❤↔ q✉②✳ √ √ ❱ỵ✐ ♠é✐ ✐ ✤➯❛♥ I t❛ ❝â Z(I) = Z( I) ✈➔ I = ∩pi ∈minAss(R/I) pi ✳ ❉♦ ✤â Z(I) = ∪Z(pi )✳ ◆➯♥ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ ❝õ❛ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ s➩ ❦❤ỉ♥❣ ✤÷đ❝ t❤➜② rã tr♦♥❣ ♣❤➙♥ t➼❝❤ ❝õ❛ t➟♣ ✤↕✐ sè t÷ì♥❣ ù♥❣✳ õ t t ỵ t ♣❤➛♥ ♥❤ó♥❣ t❤ỉ♥❣ q✉❛ ✈➼ ❞ư s❛✉✳ ✷✼ ❱➼ ❞ư ✷✳✸✳✷✶✳ ❳➨t R = K[x, y] ✈➔ I = (x2, xy)✳ ❚❛ ❜✐➳t I = (x)∩(x2, y)✳ ◆❤÷ ✈➟② t➟♣ ♥❣❤✐➺♠ ❝õ❛ I ❧➔ ❤đ♣ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ Oy ✭❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ (x)✮ ✈➔ ❣è❝ tå❛ ✤ë ✭❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ (x2 , y)✮✳ ◆❤÷♥❣ rã r➔♥❣ ❣è❝ tå❛ ✤ë ✤÷đ❝ ♥❤ó♥❣ tr♦♥❣ Oy ♥➯♥ Z(I) = Oy ❧➔ ♠ët t➟♣ ❜➜t ❦❤↔ q✉②✳ ✷✳✹✳ ■✤➯❛♥ ✤ì♥ t❤ù❝ ❚r♦♥❣ ú t tr ỵ tt t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ tr♦♥❣ ✈➔♥❤ ◆♦❡t❤❡r✳ ✷✳✹✳✶✳ P❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ t❛ ①❡♠ ①➨t ❝❤✐ t✐➳t ❤ì♥ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ R = K[x1 , , xn ] ❝õ❛ ❝→❝ ❜✐➳♥ x1 , , xn tr➯♥ tr÷í♥❣ K ✳ ❈❤♦ ✈❡❝tì ♥❣✉②➯♥ a = (a1 , , an ) ∈ Nn ✱ ❦❤✐ ✤â ♣❤➛♥ tû xa11 xann ởt ỡ tự R ỵ xa✳ ❇➟❝ ❝õ❛ ✤ì♥ t❤ù❝ xa ❧➔ |a| = a1 + + an ◆❤÷ ✈➟② ♥➳✉ xb ❧➔ ♠ët ✤ì♥ t❤ù❝ ❦❤→❝ t❤➻ xa xb = xa+b ✱ tr♦♥❣ ✤â a + b ❧➔ tê♥❣ ❝õ❛ ❤❛✐ ✈❡❝tì a ✈➔ b✳ ❈❤♦ xc ❧➔ ♠ët ✤ì♥ t❤ù❝ ❝õ❛ R s❛♦ ❝❤♦ xc = xa xb t❤➻ t❛ ♥â✐ xa ❝→❝❤ ❦❤→❝ xc ❝❤✐❛ ❤➳t ❝❤♦ xa✳ ❝❤✐❛ ❤➳t xc✱ ❤❛② ♥â✐ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✳ ■✤➯❛♥ I tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ R ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ♥➳✉ I õ ởt s ỗ t ỡ tự ❈❤ó♥❣ t❛ ❜✐➳t r➡♥❣✱ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r t❤❡♦ ✤à♥❤ ỵ ỡ s rt ộ ỡ t❤ù❝ I ✤➲✉ ❤ú✉ ❤↕♥ s✐♥❤✳ ❍ì♥ ♥ú❛✱ I ❝â ởt s ỗ ỡ tự t t tứ ởt s ỗ ỡ tự t ❦ý ❝õ❛ I ✈➔ ❝❤➾ ❣✐ú ❧↕✐ ❝→❝ ✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❜à ❝❤✐❛ ❤➳t ❜ð✐ ❝→❝ ✤ì♥ t❤ù❝ ❦❤→❝ tr♦♥❣ ❤➺ ✤â t❤➻ ❝❤ó♥❣ t❛ ❝â ❞✉② ♥❤➙t ♠ët ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ I ✳ ❚ù❝ ❧➔✱ I ❝❤➾ õ ởt t ỗ ỡ tự s ❝❤♦ ❦❤ỉ♥❣ ❝â ❤❛✐ ✤ì♥ t❤ù❝ ♥➔♦ ❝❤✐❛ ❤➳t ❝❤♦ ỵ s õ I G(I) ❈➜✉ tró❝ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✈➔ ♥❣✉②➯♥ tè r➜t ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ ❝❤➾ s✐♥❤ ❜ð✐ t➟♣ ❝→❝ ❜✐➳♥✳ ✷✽ ❇ê ✤➲ ✷✳✹✳✷✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❦❤→❝ ❦❤æ♥❣✳ ❑❤✐ ✤â✱ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ I s✐♥❤ ❜ð✐ ❝→❝ ❜✐➳♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû m ∈ G(I) ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët ❜✐➳♥ ❝õ❛ R✳ ❑❤✐ ✤â t❛ ❝â t❤➸ ♣❤➙♥ t➼❝❤ m t❤➔♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ✤ì♥ t❤ù❝ t❤ù❝ t❤ü❝ sü ❝õ❛ R ✭❦❤→❝ ❤➡♥❣✮ ❧➔ m = m1 m2 ✳ ❱➻ I ❧➔ ♥❣✉②➯♥ tè ♥➯♥ m1 ∈ I ❤♦➦❝ m2 ∈ I ✳ ❉♦ ✤â G(I) ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ I ✱ ✤✐➲✉ ♥➔② ♠➙✉ t ợ tt ữủ sỷ I s ❜ð✐ t➟♣ ❝→❝ ❜✐➳♥ t❤➻ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ I = (xr+1 , , xn )✳ ❑❤✐ ✤â✱ R/I ∼ = K[x1 , , xr ] ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ♥➯♥ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❚ø ❜ê ✤➲ tr➯♥ t❛ ❝â tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ ❝❤➾ ❝â ❤ú✉ ❤↕♥ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❧➔ ♥❣✉②➯♥ tè✳ ❚÷ì♥❣ tü✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❜➜t ❦❤↔ q✉② ♥❤÷ s❛✉✳ ❇ê ✤➲ ✷✳✹✳✸✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❦❤→❝ ❦❤ỉ♥❣✳ ❑❤✐ ✤â✱ I ❧➔ ✐✤➯❛♥ ❜➜t ❦❤↔ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â s✐♥❤ ❜ð✐ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ ❜✐➳♥✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ♠ët ♣❤÷ì♥❣ ♣❤→♣ t❤ü❝ ❤➔♥❤ ✤➸ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝✳ rữợ t ú t õ ỡ tự s ❧ô② t❤ø❛ ❝→❝ ❜✐➳♥ ❞↕♥❣ (xn1 , , xnr r ) ❧➔ (x1 , , xr )✲♥❣✉②➯♥ sì✱ ✈ỵ✐ n1 , , nr ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤♦ ♣❤➨♣ ❝❤ó♥❣ t❛ t➼❝❤ I t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❝→❝ ✐✤➯❛♥ ♥❤÷ ✈➟②✳ ▼➺♥❤ ✤➲ ✷✳✹✳✹✳ ❈❤♦ m1, , mr , u, v ❧➔ ❝→❝ ✤ì♥ t❤ù❝ ❝õ❛ R✳ ●✐↔ sû u ✈➔ v ❧➔ ❤❛✐ ✤❛ t❤ù❝ ❦❤æ♥❣ ❝â ❝❤✉♥❣ ❜✐➳♥✳ ❑❤✐ ✤â✱ (m1 , , mr , uv) = (m1 , , mr , u) ∩ (m1 , , mr , v) ❈❤ù♥❣ ♠✐♥❤✳ ❉➜✉ ❜❛♦ ❤➔♠ t❤ù❝ ” ⊆ ” ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❜❛♦ ❤➔♠ t❤ù❝ ♥❣÷đ❝ ❧↕✐✱ t❛ ❧➜② ✤ì♥ t❤ù❝ m ∈ (m1 , , mr , u) ∩ (m1 , , mr , v)✳ ❑❤✐ ✤â✱ ♥➳✉ m ❝❤✐❛ ❤➳t ❝❤♦ ♠ët tr♦♥❣ ❝→❝ ✤ì♥ t❤ù❝ mi , i = 1, r t❤➻ rã r➔♥❣ m ∈ (m1 , , mr , uv)✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥❣÷đ❝ ❧↕✐✱ t❛ s✉② r❛ m ❝❤✐❛ ✷✾ ❤➳t ❝❤♦ ❝↔ u ✈➔ v ✳ ▼➦t ❦❤→❝✱ ❞♦ u ✈➔ v ❦❤æ♥❣ ❝â ❜✐➳♥ ❝❤✉♥❣ ♥➯♥ t❛ s✉② r❛ m ❝❤✐❛ ❤➳t ❝❤♦ uv ✳ ◆❤÷ ✈➟② t❛ ❝â m ∈ (m1 , , mr , uv) ❚❤✉➟t t♦→♥ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝✿ ❈❤♦ ✐✤➯❛♥ ✤ì♥ t❤ù❝ I ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ R✳ ✣➸ t➻♠ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ I ✱ ❝❤ó♥❣ t❛ t❤ü❝ ❤✐➺♥ ❝→❝ ữợ s ữợ P t I t ❝→❝ ✐✤➯❛♥ s✐♥❤ ❜ð✐ ❧ô② t❤ø❛ ❝→❝ ❜✐➳♥ ❜➡♥❣ ❝→❝❤ t ữợ ự tr ữợ ✸✿ ◆❤â♠ ❝→❝ ✐✤➯❛♥ ❝â ❝ò♥❣ ❝➠♥ ❧↕✐ ✭❚❤❡♦ ✣à♥❤ ỵ ởt q trỹ t tứ tt t tr➯♥ ✤â ❧➔✿ ▼å✐ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✤➲✉ ❝â ♠ët t sỡ ỗ ỡ tự ❞♦ ✤â ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ❝→❝ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ✤➲✉ s✐♥❤ ❜ð✐ ❝→❝ ❜✐➳♥✳ ❱➼ ❞ư ✷✳✹✳✺✳ ❈❤♦ ✐✤➯❛♥ ✤ì♥ t❤ù❝ I = (a2, b3, ab2c) ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ R = K[a, b, c]✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✳✹ t❛ ❝â ♣❤➙♥ t➼❝❤ s❛✉✿ I = (a2 , b3 , ab2 c) = (a2 , b3 , a) ∩ (a2 , b3 , b2 c) = (a, b3 ) ∩ (a2 , b3 , b2 ) ∩ (a2 , b3 , c) = (a, b3 ) ∩ (a2 , b2 ) ∩ (a2 , b3 , c) ◆❤÷ ✈➟②✱ t❛ ❝â ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ I ❧➔✿ I = [(a, b3 ) ∩ (a2 , b2 )] ∩ (a2 , b3 , c) = (a2 , ab2 , b3 ) ∩ (a3 , b3 , c2 , a2 b2 ) ❉♦ ✤â AssR (R/I) = {(a, b), (a, b, c)} ✣è✐ ✈ỵ✐ ✐✤➯❛♥ ✤ì♥ t❤ù❝✱ ❝❤ó♥❣ t❛ ❝â ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ♥❤÷ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✹✳✻✳ ❈❤♦ ✐✤➯❛♥ ✤ì♥ t❤ù❝ I ✈➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛ R✳ ❑❤✐ ✤â p ∈ AssR(R/I) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tỗ t ởt ỡ tự m R s p = I : m ỗ t ỳ ởt ỗ t ỳ ỡ ổ ữợ G ởt ❝â t❤ù tü ❤❛✐ t➟♣ ❤ñ♣ G = (V (G) , E (G))✱ tr♦♥❣ ✤â t➟♣ V (G)) ❤ú✉ ❤↕♥ ỏ t E(G) ỗ ởt số t õ ♣❤➛♥ tû ❝õ❛ V (G) ◆❤➟♥ ①➨t ✷✳✹✳✽✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ V (G) ❣å✐ ❧➔ ✤➾♥❤✱ ❝→❝ ♣❤➛♥ tû ❝õ❛ E(G) ❣å✐ ❧➔ ❝↕♥❤✳ ◆➳✉ e = {a, b} ❧➔ ♠ët ❝↕♥❤ ❝õ❛ G t❤➻ a ✈➔ b ❣å✐ ❧➔ ❝→❝ ✤➾♥❤ ✤➛✉ ♠ót ❝õ❛ ❝↕♥❤ e ❤❛② ❝→❝ tở ợ e ữớ t tữớ ỗ t G tr t ữ s ỗ t ữủ tr t ỏ ỗ t ữủ ❞✐➵♥ ❜➡♥❣ ♠ët ✤÷í♥❣ ❝♦♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ❧✐➯♥ t❤✉ë❝✳ ❱➼ ❞ư ✷✳✹✳✾✳ ◆❣ơ ❣✐→❝ C5 = (V, E) ✈ỵ✐ V = {a, b, c, d, e} ✈➔ E = {{a, b} , {b, c} , {c, d} , {d, e} , {e, a}} õ tr ỗ t C5 ♥❤÷ s❛✉✿ ❍➻♥❤ ✷✳✶✿ ✸✶ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ t ợ ởt ỗ t ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✵✳ ❈❤♦ G = (V, G) ❧➔ ♠ët ỗ t ợ V = {x1, , xn} K ❧➔ ♠ët tr÷í♥❣✳ ❑❤✐ ✤â ✐✤➯❛♥ ❝↕♥❤ ❝õ❛ G ❧➔ I (G) = (xi xj |{xi , xj } ∈ E (G)) ⊂ R = K [x1 , , xn ] ◆❤➟♥ ①➨t ✷✳✹✳✶✶✳ ✣➾♥❤ xi ❣å✐ ❧➔ ✤➾♥❤ ❝æ ❧➟♣ ❝õ❛ G ♥➳✉ ❦❤æ♥❣ ❝â ❝↕♥❤ ♥➔♦ ❝õ❛ G ♥è✐ ✈➔♦ xi ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✐✤➯❛♥ ❝↕♥❤✱ ❝❤ó♥❣ t❛ ❝â I(G) = I (G\ {xi })✳ r ú t t ỗ t❤à ❦❤ỉ♥❣ ❝â ✤✐➸♠ ❝ỉ ❧➟♣✳ ❱➼ ❞ư ✷✳✹✳✶✷✳ ❳➨t ❝❤✉ tr➻♥❤ C5 tr♦♥❣ ❱➼ ❞ö ✷✳✹✳✾ t❛ ❝â I (C5 ) = (ab, bc, cd, de, ea) ⊂ R = K [a, b, c, d, e] ❚❛ ❝â ✤à♥❤ ♥❣❤➽❛ ✈➲ ✐✤➯❛♥ ✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ ♥❤÷ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✸✳ ✣➦t R = K[x1 , · · · , xd ]✳ ✣ì♥ t❤ù❝ xn ∈ [[R]] ữủ ổ ự ữỡ ợ i = 1, · · · , d t❛ ❝â ni ∈ {0, 1}✳ ■✤➯❛♥ ✤ì♥ t❤ù❝ J ⊆ R ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ ♥➳✉ ♥â s✐♥❤ ❜ð✐ ❝→❝ ✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✹✳ ❈❤♦ V = {v1 , , vd } ✈➔ ✤➦t R = K[x1 , , xd ]✳ ❱ỵ✐ ♠é✐ t➟♣ ❝♦♥ V ⊆ V ✱ ✤à♥❤ ♥❣❤➽❛ PV ⊆ R ❧➔ ✐✤➯❛♥ ✈➔ PV = ({xi |vi ∈ V }) R ❱➼ ❞ö ✷✳✹✳✶✺✳ ✣➦t R = K[x, y, z]✳ ❈→❝ ✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ R ❧➔ 1, x, y, z, xy, xz, yz, xyz ❚❛ ❝â (xy, yz) ❧➔ ✐✤➯❛♥ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣✳ ■✤➯❛♥ (x2 y, yz ) ❧➔ ✐✤➯❛♥ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣✳ ❱➻ ❝→❝ ✐✤➯❛♥ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ ❧➔ ✐✤➯❛♥ ❝➠♥ ♥➯♥ ♥â ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ✸✷ ▼➺♥❤ ✤➲ ✷✳✹✳✶✻✳ ❈❤♦ V = {v1, · · · , vd}✱ ✤➦t R = K[x1, · · · , xd]✳ ■✤➯❛♥ ✤ì♥ t❤ù❝ J ❧➔ ❦❤ỉ♥❣ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝â t➟♣ ❝♦♥ s❛♦ ❝❤♦ J = ni=1 PV ✳ R V1 , · · · , Vn ⊆ V ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✼✳ ▼ët i ❈❤♦ G ỗ t ợ t V = {v1 , · · · , vd }✳ ♣❤õ ✤➾♥❤ ❝õ❛ G ❧➔ t➟♣ ❝♦♥ V ⊆ V s❛♦ ❝❤♦ ♠é✐ ❝↕♥❤ vi vj tr♦♥❣ G ❤♦➦❝ vi ∈ V ❤♦➦❝ vj ∈ V ✳ P❤õ ✤➾♥❤ V ❧➔ ❝ü❝ t✐➸✉ ♥➳✉ ♥â ❦❤æ♥❣ t❤ü❝ sü ❝❤ù❛ ♠ët ♣❤õ ✤➾♥❤ ❦❤→❝ ❝õ❛ G✳ ❇ê ✤➲ s❛✉ ❝❤➾ r❛ q✉❛♥ ❤➺ ❣✐ú❛ ♣❤õ ✤➾♥❤ ✈➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥✳ ❇ê G ỗ t ợ t V ✈➔ ❝❤♦ V ⊆ V ✳ ✣➦t R = K[x1 , · · · , xd ]✱ ❦❤✐ ✤â IG ⊆ PV ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ V ❧➔ ♣❤õ ✤➾♥❤ ❝õ❛ G✳ ❚❛ ✈✐➳t V = {vi , · · · , vi } ❦❤✐ ✤â PV = (xi , · · · , xi )✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû IG ⊆ PV ✈➔ V ❧➔ ♠ët ♣❤õ ✤➾♥❤ ❝õ❛ G✳ ●å✐ vj vk = {v1 , · · · , vd } n n ❧➔ ♠ët ❝↕♥❤ ❝õ❛ G✳ ❱ỵ✐ ♠å✐ xj xk ∈ IG ⊆ PV = (xi1 , · · · , xin )✱ õ tỗ t xj xk (xim ) ợ ♠å✐ m✳ ❚ø ✤â s✉② r❛ j = im ❤♦➦❝ k = im ✈➔ vj = vim ∈ V ❤♦➦❝ vk = vim ∈ V ✳ ❱➟② V ❧➔ ♣❤õ ✤➾♥❤ ❝õ❛ G✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû V ❧➔ ♣❤õ ✤➾♥❤ ❝õ❛ G✳ P❤➛♥ tû xi xj ∈ IG ❧➔ t÷ì♥❣ ù♥❣ vi vj tr♦♥❣ G✳ ❱➻ V ❧➔ ♣❤õ ✤➾♥❤ ❝õ❛ G ♥➯♥ vi ∈ V ❤♦➦❝ vj ∈ V ✳ ❚ø ✤â s✉② r❛ xi ∈ PV ❤♦➦❝ xj ∈ PV ✳ ❱➟② xi xj ∈ PV ✳ ữợ ỵ q trồ t sỡ ỵ G ỗ t ợ t V R = K[x1 , · · · , xd ]✳ ❑❤✐ ✤â ✐✤➯❛♥ ❝↕♥❤ IG ⊆ R ❝â sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ♥❤÷ s❛✉✿ IG = PV = V = {v1 , · · · , vd } PV V ❚r♦♥❣ ✤â ❣✐❛♦ ✤➛✉ ❧➜② tr➯♥ t➜t ❝↔ ❝→❝ ♣❤õ ✤➾♥❤ ✈➔ ❣✐❛♦ s❛✉ ❧➜② tr➯♥ ♣❤õ ✤➾♥❤ ❝ü❝ t✐➸✉ ❝õ❛ G✳ ❍ì♥ ♥ú❛ ❣✐❛♦ s❛✉ ❧➔ rót ❣å♥✳ ✸✸ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ t❤➜② r➡♥❣ V PV PV ⊆ V PV ⊆ PV V ❧➔ rót ❣å♥✳ ❑❤✐ ✤â ❤✐➸♥ ♥❤✐➯♥✿ PV V V ▼ët ♣❤õ ✤➾♥❤ V ❝❤ù❛ ♣❤õ ✤➾♥❤ ❝ü❝ t✐➸✉ ❝õ❛ V ✳ ❚❛ ❝â IG ⊆ V PV t❤❡♦ ❇ê ✤➲ 2.4.18 ❉♦ ✤â ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♥❣÷đ❝ ❧↕✐ t ữ ỵ r IG ổ ự ữỡ 2.4.16 tỗ t t V1 , · · · , Vn ❝õ❛ V s❛♦ ❝❤♦ IG = n j=1 PVj ✱ ✈➔ t❛ ❝â IG ⊆ PVj ✳ ❚❤❡♦ ❇ê ✤➲ 2.4.18 Vj ❧➔ ♣❤õ ✤➾♥❤ ❝õ❛ G✱ ♥❣❤➽❛ ❧➔ n PVj ⊇ IG = j=1 ❱➼ ❞ö ✷✳✹✳✷✵✳ PV V ✣➦t R = K[x1 , x2 , x3 , x4 ] ❚➻♠ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ J = (x1 x2 , x2 x3 , x2 x4 , x3 x4 ) rữợ t t t ỗ t G ợ t➟♣ ✤➾♥❤ V = {v1 , v2 , v3 , v4 } t❤ä❛ ♠➣♥ J = IG ✳ ❚✐➳♣ t❤❡♦✱ t❛ t➻♠ ❝→❝ ♣❤õ ✤➾♥❤ ❝ü❝ t✐➸✉ ❝õ❛ G✿ {v1 , v3 , v4 }{v2 , v3 }{v2 , v4 } ố ũ t ỵ 2.4.19 t õ J = IG = (x1 , x3 , x4 ) ∩ (x2 , x3 ) ∩ (x2 , x4 ) ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ❝õ❛ J t ỗ t C5 tr ❱➼ ❞ö ✷✳✹✳✾✳ ❚❤❡♦ ✷✳✹✳✶✷ t❛ ❝â R = K[a, b, c, d, e] õ ỗ t C5 õ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ❝↕♥❤ ♥❤÷ s❛✉✿ IG = (a, b, d) ∩ (a, c, d) ∩ (a, c, e) ∩ (b, c, e) ∩ (b, d, e) ✸✹ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ♥❤÷ s❛✉✿ • ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ✣↕✐ sè ❣✐❛♦ ❤♦→♥ ♥❤÷✿ ❱➔♥❤✱ ♠ỉ✤✉♥✱ ✐✤➯❛♥ ♥❣✉②➯♥ tè✱ ữỡ õ ã P trồ t ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ổ tr ỵ ỡ s rt r ú tổ tr ỵ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ◆♦❡t❤❡r ✈➔ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ • P❤➛♥ ❝✉è✐ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ ữ r ỵ t ✈➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ✐✤➯❛♥ ✤ì♥ t❤ù❝✱ ỵ t ♥❣✉②➯♥ sì✳ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❚ü ❈÷í♥❣✱ ✣↕✐ sè ❤✐➺♥ ✤↕✐✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✵✼✳ ❬✷❪ ◆❣æ ❱✐➺t ❚r✉♥❣✱ ◆❤➟♣ ♠æ♥ ✤↕✐ sè ❣✐❛♦ ❤♦→♥ ✈➔ ❤➻♥❤ ❤å❝ ✤↕✐ sè✱ ◆❤➔ ①✉➜t ❜↔♥ ❑❤♦❛ ❤å❝ tü ♥❤✐➯♥ ✈➔ ❝æ♥❣ ♥❣❤➺✱ ✷✵✶✷✳ ❬✸❪ ❏✳ ❍❡r③♦❣ ❛♥❞ ❚✳ ❍✐❜✐✱ ❬✹❪ ❍✳ ▼❛ts✉♠✉r❛✱ ▼♦♥♦♠✐❛❧ ✐❞❡❛❧s✱ ❙♣r✐♥❣❡r✱ ✷✵✶✶✳ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ❚❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ st✉❞✐❡s ✐♥ ❛❞✈❛♥❝❞ ♠❛t❤❡♠❛t✐❝s✱ ✶✾✽✻✳ ❬✺❪ ❘✳ ❨✳ ❙❤❛r♣✱ ❙t❡♣s ✐♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✵✳ ✸✻ ... ✣à♥❤ ♥❣❤➽❛ ✶✳✵✳✼✳ ▼ët ✐✤➯❛♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ ♥➳✉ I = R ✈➔ ♥â ❦❤æ♥❣ ❝❤ù❛ tr t ý ởt tỹ sỹ ỵ ❤✐➺✉ Max(R) ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R ◆❤➟♥ ①➨t ✶✳✵✳✽✳ ✭✐✮ ▼å✐ ✐✤➯❛♥... ✤➲ ❩♦r♥ tỗ t tố ❈❤♦ S ❧➔ t➟♣ ❝♦♥ ❝õ❛ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ R✱ ❦❤✐ ✤â ♥➳✉ ❤❛✐ ♣❤➛♥ tû s, t ❜➜t ? ?ý t❤✉ë❝ S ♠➔ t➼❝❤ ❝õ❛ ❝❤ó♥❣ st ❝ơ♥❣ t❤✉ë❝ S t❤➻ S ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ✤â♥❣ ♥❤➙♥✳ ❚❛ q✉② ÷ỵ❝ ∈ S ✳ ❱➼... ❝♦♥ ♥❣✉②➯♥ sì✳ ▼ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ♥➳✉ t❛ ❦❤ỉ♥❣ t❤➸ ❜ä ❜➜t ? ?ý ♠ët ♠ỉ✤✉♥ ♥❣✉②➯♥ ♥➔♦ tr♦♥❣ ♣❤➙♥ t➼❝❤ ✤â✱ tù❝ ❧➔ N = N1 ∩ · · · ∩ Ni−1 ∩ Ni+1 · · · ∩ Nr ợ i =