TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ====== NGUYỄN THỊ KIM ANH VỀ MƠĐUN CĨ ĐỘ DÀI HỮU HẠN TRÊN VÀNH GIAO HỐN KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số HÀ NỘI, 2019 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ====== NGUYỄN THỊ KIM ANH VỀ MƠĐUN CĨ ĐỘ DÀI HỮU HẠN TRÊN VÀNH GIAO HỐN KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Người hướng dẫn khoa học TS Nguyễn Thị Kiều Nga HÀ NỘI, 2019 ▲❮■ ❈❷▼ ❒◆ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ❞➔✐ ♥❣❤✐➯♠ tó❝✱ ♠✐➺t ♠➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❝→❝ ❚❤➛② ❈æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✱ ✤➳♥ ♥❛② ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤✱ s➙✉ s➢❝ tỵ✐ ❝→❝ ❚❤➛② ❈æ ❣✐→♦ tr♦♥❣ tê ✣↕✐ sè✱ ❝→❝ ❚❤➛② ❈æ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ ❚✳❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✲ ♥❣÷í✐ ✤➣ trü❝ t✐➳♣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥✱ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❝❤➾ ❜↔♦ ❝❤♦ ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉✱ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❧✉➟♥✳ ▼➦❝ ❞ò ✤➣ r➜t ❝è ❣➢♥❣ ①♦♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ❝ơ♥❣ ♥❤÷ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt ữủ sỹ õ ỵ tứ ❈æ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❑✐♠ ❆♥❤ ▲❮■ ❈❆▼ ✣❖❆◆ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✧❱➲ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❞♦ sü ❝è ❣➢♥❣ ♥é ❧ü❝ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❝æ ❣✐→♦ ✲ ❚✳❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛✳ ❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ♥❤÷ ✤➣ ✈✐➳t tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❱➻ ✈➟②✱ ❡♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝õ❛ r✐➯♥❣ ❡♠✱ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❜➜t ❦➻ ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ♥➔♦ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❑✐♠ ❆♥❤ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼æ✤✉♥ ❝♦♥ ✳ ✳ ✳ ✳ ổ tữỡ ỗ ♠æ✤✉♥ ✳ ✳ ✳ ❚ê♥❣ ✈➔ t➼❝❤ trü❝ t✐➳♣ ✳ ❉➣② ❦❤ỵ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✣ë ❞➔✐ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶ ▼æ✤✉♥ ◆♦❡t❤❡r ✈➔ ♠æ✤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ▼æ✤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ▼æ✤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣ë ❞➔✐ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ▼ỉ✤✉♥ ✤ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❉➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ ổ ỵ r ❍♦❧❞❡r ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✹ ✣à♥❤ ♥❣❤➽❛ ✭✣ë ❞➔✐ ❝õ❛ ♠ỉ✤✉♥✮ ✳ ✷✳✸ ✣➦❝ tr÷♥❣ ❝õ❛ ♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✺ ✳ ✻ ✳ ✼ ✳ ✶✵ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✾ ✶✾ ✷✹ ✸✵ ✸✵ ✸✶ ✸✶ ✸✺ ✸✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✷✳✸✳✶ ❚➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ ✤ë ❞➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸✳✷ ✣✐➲✉ ❦✐➺♥ ✤➸ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸✳✸ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✸✽ ❑➳t ❧✉➟♥ ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✐✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ▲í✐ ♠ð ✤➛✉ ✣↕✐ sè ❤✐➺♥ ✤↕✐ ✤÷đ❝ ①➙② ❞ü♥❣ tø ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ❝ì ❜↔♥✿ ♥❤â♠✱ ✈➔♥❤✱ tr÷í♥❣✱✳ ✳ ✳ ❚r♦♥❣ ✤â ❝➜✉ tró❝ ♠ỉ✤✉♥ ❝â ✈❛✐ trá r➜t q✉❛♥ trå♥❣✳ ◆â ❝â ❦❤↔ ♥➠♥❣ t❤è♥❣ ♥❤➜t ♠ët ❝→❝❤ ❜↔♥ ❝❤➜t ❝→❝ ❝➜✉ tró❝ ✈➔♥❤✱ ✐✤❡❛♥✱ ♥❤â♠ ❆❜❡♥✱ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì✳ ✳ ✳ ❚➼♥❤ ❧✐♥❤ ❤♦↕t ✈➔ tê♥❣ q✉→t ❝õ❛ ❝➜✉ tró❝ ♠ỉ✤✉♥ ✤❡♠ ❧↕✐ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ t♦ ❧ỵ♥ tr♦♥❣ ✤↕✐ sè✳ ❚r♦♥❣ ❝➜✉ tró❝ ♠ỉ✤✉♥✱ ✏✣ë ❞➔✐ ♠ỉ✤✉♥✑ ❧➔ ✈➜♥ ữủ ữớ q t ợ ố t ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❝➜✉ tró❝ ♠ỉ✤✉♥ ✈➔ ✤÷đ❝ sü ữợ ổ ❝❤å♥ ✤➲ t➔✐ ✏❱➲ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✑ ❧➔♠ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳ ◆ë✐ ❞✉♥❣ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ♥❤➜t ❝õ❛ ♠ỉ✤✉♥✱ ✤➦❝ ❜✐➺t ❧➔ ❦❤→✐ ♥✐➺♠ ✧▼æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✧✳ ◆ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥ ❝❤✐❛ ❧➔♠ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r➻♥❤ ❜➔② ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ✈➲ ♠æ✤✉♥✱ ♠æ✤✉♥ ❝♦♥✱ ổ tữỡ ỗ ổ tờ t trỹ t ợ ỵ q ợ ❝❤➫ r❛✳ ❈❤÷ì♥❣ ✷✿ ✣ë ❞➔✐ ♠ỉ✤✉♥ ❚r➻♥❤ ❜➔② ♠ët sè ♠ỉ✤✉♥ ✤➦❝ ❜✐➺t✿ ♠ỉ✤✉♥ ✤ì♥✱ ♠ỉ✤✉♥ ◆♦❡t❤❡r✱ ♠ỉ✤✉♥ ❆rt✐♥✱ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❞➣② ❤ñ♣ t❤➔♥❤ ✈➔ ✤ë ❞➔✐ ❝õ❛ ♠æ✤✉♥✳ ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ❝â ❤↕♥ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ❝â ♥❤ú♥❣ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ữủ ỵ õ õ ổ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❑✐♠ ❆♥❤ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♠æ✤✉♥ ✤➸ ❝❤✉➞♥ ❜à ❝❤♦ ❝❤÷ì♥❣ s❛✉✳ ✶✳✶ ▼ỉ✤✉♥ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ R ❧➔ ✈➔♥❤ ❝â ✤ì♥ ✈à ❦➼ ❤✐➺✉ ❧➔ 1✳ ▼ët t➟♣ ❤đ♣ M ✤÷đ❝ ❣å✐ ❧➔ ♠ët R ✲ ♠ỉ✤✉♥ tr→✐ ❤❛② ❝á♥ ❣å✐ ❧➔ ♠æ✤✉♥ tr→✐ tr➯♥ ✈➔♥❤ R ♥➳✉ tr➯♥ M ❝â ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ✈æ ữợ ợ tỷ R ữủ M ×M →M (m, m ) → m + m • R×M →M (r, m) → rm t❤ä❛ ♠➣♥✿ ✭✐✮ M ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ❧➔ ♠ët ♥❤â♠ ❆❜❡❧✳ ✭✐✐✮ M ợ ổ ữợ tọ t t s ợ tỷ tũ ỵ m, m M ✈➔ r, r ∈ R✿ ✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ • r(m + m ) = rm + rm ❀ • (r + r )m = rm + r m❀ • (rr )m = r(r m) 1.m = m ữỡ tü t❛ ✤à♥❤ ♥❣❤➽❛ R ✲ ♠ỉ✤✉♥ ♣❤↔✐ ✈ỵ✐ ❝→❝ tỷ R tr ổ ữợ ❜➯♥ ♣❤↔✐✳ ◆❤➟♥ ①➨t✿ ◆➳✉ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ t❤➻ ❦❤→✐ ♥✐➺♠ ♠ỉ✤✉♥ tr→✐ ✈➔ ♠ỉ✤✉♥ ♣❤↔✐ trò♥❣ ♥❤❛✉✳ ❱➻ ✈➟② ✤➸ t❤✉➟♥ t✐➺♥ t❛ ❝❤➾ ①➨t ❝→❝ R ✲ ♠æ✤✉♥ tr→✐ ✈➔ ❣å✐ t➢t ❧➔ R ✲ ♠æ✤✉♥✳ ✶✳✶✳✷ ❱➼ ❞ư ❱➼ ❞ư ✶✳ ▼é✐ V ✲ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ K ❧➔ ♠ët K ✲ ♠ỉ✤✉♥✳ ❱➼ ❞ö ✷✳ ▼é✐ ♥❤â♠ ❆❜❡❧ ❝ë♥❣ M ✤➲✉ ❧➔ Z ổ ợ ổ ữợ ữủ ữ s ợ m M a Z t❛ ❝â✿ m + m + + m a am = (−m) + (−m) + + (−m) ♥➳✉ a > ♥➳✉ a = ♥➳✉ a < |a| ❱➼ ❞ö ợ ởt õ G trữợ t E = End(G, G) t tỹ ỗ ♥❤â♠ G✳ ❙✉② r❛ E = End(G, G) ❧➔ ♠ët ✈➔♥❤ ❝â ✤ì♥ ✈à ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ①→❝ ✤à♥❤ ❜ð✐✿ ❱ỵ✐ ♠å✐ f, g ∈ E ✱ ✈ỵ✐ ♠å✐ x ∈ G t❤➻ (f + g)(x) = f (x) + g(x) (f g)(x) = f (g(x)) ✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ n1 → (n1 , 0) g : N1 ⊕ N2 → N2 ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝❤➼♥❤ t➢❝ (n1 , n2 ) → n2 ❉♦ ✤â N1 ⊕ N2 ❧➔ ♠æ✤✉♥ ❆rt✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ N1, N2 ❧➔ ♠ỉ✤✉♥ ❆rt✐♥✳ k ●✐↔ sû ✤✐➲✉ ♥➔② ✤ó♥❣ ✈ỵ✐ m = k tù❝ ❧➔ i=1 ⊕ Ni ❧➔ ♠æ✤✉♥ ❆rt✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Ni ❧➔ ♠ỉ✤✉♥ ❆rt✐♥✱ ✈ỵ✐ ♠å✐ i = 1, k ❚❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ m = k + t❤➻ ✤✐➲✉ ♥➔② ❝ơ♥❣ ✤ó♥❣✳ k+1 k ❚❤➟t ✈➟②✿ i=1 ⊕ Ni = ⊕ Ni + Nk+1 i=1 ❉♦ ✤ó♥❣ ✈ỵ✐ m = ♥➯♥ i=1 ⊕ Ni ❧➔ ♠æ✤✉♥ ❆rrt✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Ni ❧➔ ♠ỉ✤✉♥ ❆rt✐♥✱ ✈ỵ✐ ♠å✐ i = 1, k✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ k+1 ❍➺ q✉↔ ✷✳ ✭✐✮ ❈→❝ ♠ỉ✤✉♥ ❝♦♥✱ ♠ỉ✤✉♥ t❤÷ì♥❣ ❝õ❛ ♠ët ♠ỉ✤✉♥ rt tữỡ ự ụ ổ rt ỗ ❝➜✉ ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❉ü❛ ✈➔♦ ♠➺♥❤ ✤➲ ✷✳✶✳✷✳✸ ✭✐✐✮ ●✐↔ sû M ❧➔ R ✲ ♠æ✤✉♥ ❆rt✐♥✳ ●✐↔ sû t❛ ❝â f : M M R ỗ ổ ✤â t❛ ❝â f (M ) = Imf M/ker f ❚❤❡♦ ✭✐✮ t❤➻ M/Kerf ❧➔ ♠æ✤✉♥ ❆rt✐♥ ♥➯♥ f (M ) ❧➔ ♠ỉ✤✉♥ ❆rt✐♥✳ ✷✳✶✳✷✳✺ ❱➼ ❞ư ❱➼ ❞ư ✶✳ ❈❤♦ R ❧➔ ♠ët tr÷í♥❣ t❤➻ R ❧➔ R ✲ ♠æ✤✉♥✳ ❑❤✐ ✤â R ❧➔ R ✲ ♠æ✤✉♥ ❆rt✐♥✳ ❱➼ Z ổ rt tỗ t↕✐ ❞➣② ❣✐↔♠ ❝→❝ ✐✤➯❛♥ ✷✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ s❛✉ ❝õ❛ Z ❦❤æ♥❣ ❞ø♥❣ 2Z ⊃ 22 Z ⊃ 2k Z ⊃ ❱➼ ❞ö ✸✳ ●✐↔ sû A ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ tr➯♥ tr÷í♥❣ K ✳ ❑❤✐ ✤â A ❧➔ K ✲ ♠æ✤✉♥ ❆rt✐♥✳ ✷✳✶✳✷✳✻ ▼➺♥❤ ✤➲ M ❈❤♦ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❧➔ ♠ỉ✤✉♥ ❆rt✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❈❤ù♥❣ ♠✐♥❤✳ ⇒ • N ✈➔ M/ N R ✲ ♠æ✤✉♥ M✳ ❑❤✐ ✤â ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ●✐↔ sû M ❧➔ R ✲ ♠æ✤✉♥ ❆rt✐♥✳ ❳➨t ♠å✐ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ N N1 ⊃ N2 ⊃ ⊃ Nn ⊃ ❝ô♥❣ ❧➔ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❉♦ M ổ rt tỗ t n N ✤➸ Nn = Nn+1 = ❱➟② N ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ • ●✐↔ sû M1 ⊃ M2 ⊃ ⊃ Mn ⊃ ❧➔ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M/N ✳ ❚r♦♥❣ ✤â ♠æ✤✉♥ ❝♦♥ ❝õ❛ M/N ❝â ❞↕♥❣ P/N ✈ỵ✐ P ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ✈➔ P/N M/N õ tỗ t ♠æ✤✉♥ ❝♦♥ ❝õ❛ M N1 ⊃ N2 ⊃ ⊃ Nn ⊃ s❛♦ ❝❤♦ Mi = Ni/N ❉♦ M ổ rt tr ứ õ tỗ t↕✐ n ∈ N ✤➸ Nn = Nn+1 = ự tỗ t n N Nn/N = Nn+1/N = ❍❛② Mn = Mn+1 = ❱➟② M/N ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ✷✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ◆➳✉ N ✈➔ M/N ❧➔ ♠ỉ✤✉♥ ❆rt✐♥✳ ❱ỵ✐ ♠å✐ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M ✿ M1 ⊃ M2 ⊃ ⊃ Mn ⊃ ❚❛ ❝â ❞➣② ❣✐↔♠ t÷ì♥❣ ù♥❣ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ N ❧➔ ⇐ M1 ∩ N ⊃ M2 ∩ N ⊃ ⊃ Mn ∩ N ⊃ ❱➔ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M/N ❧➔ M1 + N/ ⊃ M2 + N/ ⊃ M3 + N/ ⊃ N N N ❉♦ M/N ✈➔ N ❧➔ ♠æ✤✉♥ ❆rt✐♥ ♥➯♥ ❤❛✐ ❞➣② tr➯♥ ứ tự tỗ t n N Mn ∩ N = Mn+1 ∩ N = Mn + N/ ⊃ Mn+1 + N/ ⊃ N N Mn + N = Mn+1 + N = ❙✉② r❛ ❱ỵ✐ ♠å✐ i ≥ n t❛ ❝â✿ Mi = Mi ∩ (Mi + N ) = Mi ∩ (Mi+1 + N ) = Mi+1 + (Mi ∩ N ) = Mi+1 + (Mi+1 ∩ N ) = Mi+1 t ợ x Mi+1 + (Mi ∩ N )✱ ✤➦t x = u + v tr♦♥❣ ✤â u ∈ Mi+1 ✱ v ∈ Mi ∩ N ✳ ❉♦ Mi ⊃ Mi+1 ♥➯♥ u ∈ Mi v ∈ M ,v ∈ N i ❙✉② r❛ x = u + v ∈ Mi x = u + v ∈ Mi+1 + N x ∈ Mi ∩ (Mi+1 + N ) ❍❛② ❱➟② Mi+1 + (Mi ∩ N ) ⊂ Mi ∩ (Mi+1 + N ) ữủ ✈ỵ✐ ♠å✐ x ∈ Mi ∩ (Mi+1 + N ) t❤➻ ✷✾ x ∈ Mi x ∈ Mi+1 + N ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣➦t x = u + v ✈ỵ✐ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ x ∈ M ⇒ u, v ∈ M i i x ∈ M ,v ∈ N i+1 ❚❤➻ v = x − u ∈ Mi ✭ ✈➻ x ∈ Mi, u ∈ Mi+1 ⊂ Mi✮ ❚ø v ∈ Mi, v ∈ N s✉② r❛ v ∈ Mi ∩ N ✳ ❚ø ✤â t❛ s✉② r❛ x = u+v ∈ Mi+1 + (Mi ∩ N ) ❱➟② Mi ∩ (Mi+1 + N ) ⊂ Mi+1 + (Mi ∩ N ) ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮ s✉② r❛ Mi = Mi ∩ (Mi + N ) = Mi ∩ (Mi+1 + N ) = Mi+1 + (Mi ∩ N ) = Mi+1 + (Mi+1 ∩ N ) = Mi+1 ❚ù❝ Mi = Mi+1 ✈ỵ✐ ♠å✐ i ≥ n ❱➟② M ❧➔ R ✲ ♠æ✤✉♥ ❆rt✐♥✳ ✷✳✷ ✣ë ❞➔✐ ♠ỉ✤✉♥ ✷✳✷✳✶ ▼ỉ✤✉♥ ✤ì♥ ✷✳✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ M ❧➔ ♠ët R ✲ ♠ỉ✤✉♥✳ M ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➳✉ M ❝❤➾ ❝â ❞✉② ♥❤➜t ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❧➔ ✈➔ ❝❤➼♥❤ ♥â✳ ✷✳✷✳✶✳✷ ❱➼ ❞ö ❱➼ ❞ö ✶✳ ❈❤♦ V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ♠ët ❝❤✐➲✉ tr➯♥ K ✳ ❑❤✐ ✤â V ❧➔ ♠ët K ✲ ♠æ✤✉♥ ✤ì♥✳ ❱➼ ❞ư ✷✳ ▼ët ♥❤â♠ ❝ë♥❣ ❆❜❡❧ ❝➜♣ ♥❣✉②➯♥ tè ❧➔ ♠ët Z ✲ ♠ỉ✤✉♥ ✤ì♥✳ ✸✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✷✳✷✳✷ ❉➣② ❤đ♣ t❤➔♥❤ ❝õ❛ ♠ỉ✤✉♥ ✷✳✷✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ▼ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ ♠ët R ✲ ♠ỉ✤✉♥ A ❧➔ ♠ët ❞➣② ❤ú✉ ❤↕♥ ❣✐↔♠ ❞➛♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✿ A = A0 ⊃ A1 ⊃ ⊃ Ak = {0} s❛♦ ❝❤♦ ♠é✐ Ai/Ai+1 ❧➔ ♠ët ♠ỉ✤✉♥ ✤ì♥✱ ✈ỵ✐ ♠å✐ i = 0, k − ❑❤✐ ✤â k ❣å✐ ❧➔ ✤ë ❞➔✐ ❝õ❛ ❞➣② ❤ñ♣ t❤➔♥❤✳ ✷✳✷✳✷✳✷ ❱➼ ❞ư ❱➼ ❞ư ✶✳ ❈❤♦ V ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ tỡ ỳ tr ởt trữớ K ợ ❝ì sð ❧➔ {x1, x2, , xk }✳ k ❑❤✐ ✤â {0} ⊂ a1K ⊂ a1K + a2K ⊂ ⊂ aiK = V ❧➔ ♠ët ❞➣② ❤ñ♣ i=1 t❤➔♥❤ ❝õ❛ V ❝â ✤ë ❞➔✐ ❧➔ k✳ ❱➼ ❞ö ✷✳ ❳➨t ✈➔♥❤ sè ♥❣✉②➯♥ Z ♥❤÷ ❧➔ ♠ỉ✤✉♥ tr➯♥ ❝❤➼♥❤ ♥â✳❚❛ ❝â A ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ Z ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = nZ, n ∈ Z✳ ●✐↔ sû Z ❝â ❞➣② ❤ñ♣ t❤➔♥❤ ❧➔ Z = A0 ⊃ A1 ⊃ ⊃ Am = ●✐↔ sû Am−1 = kZ✳ ❚❛ ❧✉ỉ♥ t➻♠ ✤÷đ❝ ♠ët ✐✤➯❛♥ ♥➡♠ ❣✐ú❛ ✵ ✈➔ Am−1✳ ❈❤➥♥❣ ❤↕♥ Am−1 ⊃ k2Z ⊃ 0✳ ❱➟② t❛ s✉② r❛ Z ❦❤ỉ♥❣ ❝â ❞➣② ❤đ♣ t❤➔♥❤✳ ❱➼ ❞ư ✸✳ ▼ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ❝â ❞➣② ❤đ♣ t❤➔♥❤ ✈ỵ✐ ✤ë ❞➔✐ ❧➔ k ❦❤✐ ✈➔ ❝❤➾ õ õ k ỵ r ✲ ❍♦❧❞❡r ●✐↔ sû R ✲ ♠æ✤✉♥ A ❝â ❞➣② ❤ñ♣ t❤➔♥❤✱ ❦❤✐ ✤â t❛ ❦➼ ❤✐➺✉ l(A) ❧➔ ✤ë ❞➔✐ ❝õ❛ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ ♥❤ä ♥❤➜t ❝õ❛ A✳ ❚❛ ❝â ❜ê ✤➲ s❛✉ ✤➙②✿ ✷✳✷✳✸✳✶ ❇ê ✤➲ ●✐↔ sû R ✲ ♠æ✤✉♥ A ❝â ❞➣② ❤ñ♣ t❤➔♥❤ ✈➔ N ❧➔ ♠ët ✸✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ♠æ✤✉♥ ❝♦♥ ❝õ❛ A✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✭✐✮ N ❝ơ♥❣ ❝â ❞➣② ❤đ♣ t❤➔♥❤ ✈ỵ✐ l(N ) ≤ l(A)✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ N = A❀ ✭✐✐✮ ▼ỉ✤✉♥ t❤÷ì♥❣ A/N ❝ơ♥❣ ❝â ❞➣② ❤đ♣ t❤➔♥❤ ✈ỵ✐ l(A/N ) ≤ l(A)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû A ❝â ❞➣② ❤ñ♣ t❤➔♥❤ A = A0 ⊃ A1 ⊃ ⊃ An = {0} ✈ỵ✐ ✤ë ❞➔✐ ♥❤ä ♥❤➜t l(A) = n✳ ❑❤✐ ✤â t❛ ❝â N = N ∩ A0 ⊇ N ∩ A1 ⊇ ⊇ N ∩ An = {0} ✭✶✮ tr ữủ r tữỡ ự f : N ∩ Ai−1/N ∩ Ai → Ai−1/Ai x + N ∩ Ai → x + Ai ❧➔ ♠ët ✤ì♥ ❝➜✉✳ ◆❤ỵ ❧↕✐ r➡♥❣ Ai−1/Ai ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➯♥ N ∩ Ai−1/N ∩ Ai ❤♦➦❝ ❧➔ ♠ỉ✤✉♥ ❦❤ỉ♥❣ ✭ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ N ∩ Ai−1 = N ∩ Ai✮ ❤♦➦❝ ❧➔ ợ Ai1/Ai ởt ổ ỡ ữ ❜➡♥❣ ❝→❝❤ ❧÷đ❝ ❜ä ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ tr♦♥❣ ❞➣② ✭✶✮ t❛ ✤÷đ❝ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ N ợ ổ ữủt q l(A) = n ứ ✤â s✉② r❛ l(N ) ≤ l(A)✳ ◆❤➟♥ t❤➜② r➡♥❣ l(N ) = l(A) t❤➻ ✭✶✮ ♣❤↔✐ ❧➔ ♠ët ❞➣② ủ t tữỡ ữỡ ợ N Ai1/N ∩ Ai ∼ = Ai−1/Ai ✱ ✈ỵ✐ ♠å✐ i = 1, n✳ ❚ø N ∩ An−1/N ∩ An ∼ = An1/An ợ ú ỵ r An = {0} t rút r❛ N ∩ An−1 = An−1 ✳ ❑➳t q✉↔ ♥➔② ❞➝♥ ✤➳♥ ✤➥♥❣ t❤ù❝ N ∩ An−2/An−1 ∼ = An−2/ An−1 ✈➔ ❞♦ ✤â N ∩ An−2 = An−2 ✳ ▲➦♣ ❧↕✐ ❧➟♣ ❧✉➟♥ ✈ø❛ sû ❞ö♥❣ t❛ s✉② r❛ N ∩ Ai = Ai✱ ✈ỵ✐ ♠å✐ i = 0, n✳ ✣➦❝ ❜✐➺t N = N ∩ A = N ∩ A0 = A0 = A✳ ◆❤÷ ✈➟② ✭✐✮ ✤➣ ữủ ự t tứ ủ t ❜❛♥ ✤➛✉ ❝õ❛ A t❛ ♥❤➟♥ ✤÷đ❝ ❞➣② ✸✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ A/ = A0 + N/ ⊇ A1 + N/ ⊇ ⊇ An + N/ = {0} N N N N ❚❤❡♦ ❤➺ q✉↔ ✸ ♣❤➛♥ ✶✳✹✳✽ t❛ ❝â ✭✷✮ Ai−1 + N/ A A +N ∼ N ∼ ✱ = Ai +Ni−1 = Ai−1 ∩Ai−1 ✈ỵ✐ ♠å✐ i = 1, n i +N Ai + N/ N ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ Ai−1/Ai + N ∩ Ai−1 ❧➔ ♠ët ♠ỉ✤✉♥ t❤÷ì♥❣ ❝õ❛ ♠ỉ✤✉♥ ✤ì♥ Ai−1/Ai ♥➯♥ ♥â ❤♦➦❝ ❧➔ ♠ỉ✤✉♥ ❦❤ỉ♥❣ ❤♦➦❝ ❧➔ ♠ỉ✤✉♥ ✤ì♥✳ ❇➡♥❣ ✈✐➺❝ ❧÷đ❝ ❜ä ❝→❝ t❤➔♥❤ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❝õ❛ ✭✷✮ t❛ t❤✉ ✤÷đ❝ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ A/N ❝â ✤ë ❞➔✐ ❦❤ỉ♥❣ ✈÷đt q✉→ l(A) = n✳ ❱➟② l(A/N ) ≤ l(A) r ữủ ự ỵ ❏♦r❞❛♥ ✲ ❍♦❧❞❡r ❈❤♦ A ❧➔ ♠ët R ✲ ♠æ✤✉♥ ❝â ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ ❝ơ♥❣ ❝â ✤ë ❞➔✐ k k✳ ❑❤✐ ✤â ❝â ♠å✐ ❞➣② ❤ñ♣ t❤➔♥❤ ❦❤→❝ ❝õ❛ ✈➔ ♠ët ❞➣② ❣✐↔♠ ❞➛♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❦➨♦ ❞➔✐ tỵ✐ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ ❈❤ù♥❣ ♠✐♥❤✳ A A ❝â t❤➸ A✳ ❑➼ ❤✐➺✉ l(A) ❧➔ ✤ë ❞➔✐ ❞➣② ❤ñ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ ♥❤ä ♥❤➜t ❝õ❛ A✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤✿ • ◆➳✉ B ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ A t❤➻ l(B) ≤ l(A)✳ • ◆➳✉ B ❧➔ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ A t❤➻ l(B) < l(A) ❚❤➟t ✈➟②✱ ❣✐↔ sû A = A0 ⊃ A1 ⊃ ⊃ An t❤➔♥❤ ❝õ❛ A ❝â n = l(A) ❧➔ ✤ë ❞➔✐ ♥❤ä ♥❤➜t✳ ❑❤✐ ✤â ✈ỵ✐ B ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ A B = (B ∩ A0 ) ⊃ (B ∩ A1 ) ⊃ ⊃ (B ∩ An ) = ❞➛♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ B ✳ ❱ỵ✐ i = 0, n − 1✱ t❛ ❝â →♥❤ ①↕✿ f : (B ∩ Ai )/(B ∩ Ai+1 ) → Ai/Ai+1 = ✭✯✮ ❧➔ ♠ët ❞➣② ❣✐↔♠ ❧➔ ♠ët ✤ì♥ →♥❤✳ ✸✸ ❧➔ ♠ët ❞➣② ❤ñ♣ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❉♦ Ai/Ai+1 ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➯♥ (B ∩ Ai)/(B ∩ Ai+1) ❧➔ ♠ỉ✤✉♥ ✤ì♥✳ ❙✉② r❛ (B ∩ Ai) = (B ∩ Ai+1) ❤♦➦❝ B ∩ Ai+1 = ❇➡♥❣ ❝→❝❤ ❜ä ✤✐ ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥ ✤÷đ❝ ❧➦♣ ❧↕✐ tr♦♥❣ ❞➣② ✭✯✮✳ ✭❈❤➥♥❣ ❤↕♥ ♥➳✉ ❝â (B ∩ Ai) = (B ∩ Ai+1) t❛ s➩ ❜ä (B ∩ Ai+1)✮✳ ❚ø ✤➙② t❛ ✤÷đ❝ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ B ❝â ✤ë ❞➔✐ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ n✳ ❚ø ✤â l(B) ≤ n = l(A)✳ ●✐↔ sû l(B) = l(A)✳ ❑❤✐ ✤â ✈ỵ✐ ≤ i ≤ n − t❤➻ (B ∩ Ai)/(B ∩ Ai+1) = ✤➥♥❣ ❝➜✉ ✈ỵ✐ Ai/ Ai+1 ✳ ❉♦ An = ♥➯♥ B ∩ An−1/B ∩ {0} = An−1/{0} ❙✉② r❛ B ∩ An−1 = An−1 ❚÷ì♥❣ tü B ∩ An−2 = An−2✱✳✳✳✱ B ∩ A0 = A0 ❚ù❝ ❧➔ A = B ✳ ❉♦ ✤â ♥➳✉ B ❧➔ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ A t❤➻ l(B) < l(A)✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♠å✐ ❞➣② ❤ñ♣ t❤➔♥❤ ✤➲✉ ❝â ❝ò♥❣ ✤ë ❞➔✐✳ ❚❤➟t ✈➟②✱ ①➨t ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ ❆ ❝â ✤ë ❞➔✐ t ❧➔ A = A0 ⊃ A1 ⊃ ⊃ At = ❚❤❡♦ tr➯♥ t❛ ❝â✿ n = l(A) > l(A1) > > l(At) = ❙✉② r❛ n ≥ t✳ ✭✶✮ ▼➦t ❦❤→❝ n ❧➔ ✤ë ❞➔✐ ♥❤ä ♥❤➜t ❝õ❛ ❞➣② ❤ñ♣ t❤➔♥❤ A ♥➯♥ n ≤ t✭✷✮✳ ❚ø ✭✶✮ ✈➔ ✭✷✮ t❛ ❝â n = t✳ ❑❤✐ ✤â ♠å✐ ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ ▼ ✤➲✉ ❝â ❝ò♥❣ ✤ë ❞➔✐ n = l(A)✳ ❚✐➳♣ t❤❡♦✱ ❣✐↔ sû A = A0 ⊃= A1 ⊃= ⊃= Ab = ✭✯✯✮ ❧➔ ♠ët ❞➣② ❣✐↔♠ ❞➛♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❆✳ ✸✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ◆➳✉ b < l(A) t❤➻ ✭✯✯✮ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ủ t õ tỗ t i ∈ {0, 1, , b − 1} ✤➸ Ai/Ai+1 ❦❤æ♥❣ ổ ỡ tỗ t ổ N s❛♦ ❝❤♦ Ai ⊃= N ⊃= Ai+1✳ ❚❛ t❤➯♠ N ữủ ởt ợ A = A0 ⊃ A1 ⊃ ⊃ Ai ⊃ N ⊃ Ai+1 ⊃ ⊃ Ab = = = = = = = = ◆➳✉ b + < l(A) t❛ t✐➳♣ tö❝ ❧➦♣ ❧↕✐ q✉→ tr➻♥❤ tr➯♥✳ ❱➟② s ởt số ữợ t ữủ ởt ủ t ❝õ❛ ❆✳ ✷✳✷✳✹ ✣à♥❤ ♥❣❤➽❛ ✭✣ë ❞➔✐ ❝õ❛ ♠æ✤✉♥✮ ✷✳✷✳✹✳✶ ✣à♥❤ ♥❣❤➽❛ ◆➳✉ ♠ët R ✲ ♠æ✤✉♥ A ❝â ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ t❤➻ ✤ë ❞➔✐ ❝õ❛ ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ A ❣å✐ ❧➔ ✤ë ❞➔✐ ❝õ❛ ♠æ✤✉♥ A✳ ❑➼ ❤✐➺✉ ❧➔ lR(A) ❤♦➦❝ l(A)✳ ◆➳✉ l(A) < ∞ t❤➻ t❛ ♥â✐ ♠æ✤✉♥ ❆ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ◆➳✉ ổ õ ủ t t t q ữợ l(A) = ∞ ✈➔ ❣å✐ ♥â ❧➔ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ✈ỉ ❤↕♥✳ ✷✳✷✳✹✳✷ ❱➼ ❞ư ❱➼ ❞ư ✶✳ ✣ë ❞➔✐ ❝õ❛ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ❝❤➼♥❤ ❧➔ sè ❝❤✐➲✉ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ✤â✳ ❱➼ ❞ư ✷✳❱ỵ✐ p, q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â lZ(Zpq ) = 2✱ ✈➻ Zpq ❝â ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❧➔ Zpq ⊃ pZpq ⊃ {0}✳ ð ✤â pZpq = {pa|a ∈ Zpq }✳ ❱➼ ❞ö ✸✳ Z ✲ ♠ỉ✤✉♥ Z6 = Z/6Z ❝â ❞➣② ❤đ♣ t❤➔♥❤ ⊂ 2Z/6Z ⊂ Z/6Z ❝â ✤ë ❞➔✐ l(Z6) = ✸✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✷✳✸ ✣➦❝ tr÷♥❣ ❝õ❛ ♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✷✳✸✳✶ ❚➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ ✤ë ỵ M f M →g M ❝→❝ ❘ ✲ ♠æ✤✉♥✳ ❑❤✐ ✤â ❈❤ù♥❣ ♠✐♥❤✳ → ❧➔ ♠ët ❞➣② ❦❤ỵ♣ ♥❣➢♥ l(M ) = l(M ) + l(M ) ỵ tr ổ ú ♥➳✉ ♠ët tr♦♥❣ ❝→❝ ♠æ✤✉♥ tr➯♥ ❝â ✤ë ❞➔✐ ∞✳ ●✐↔ sû t➜t ❝↔ ❝→❝ ♠æ✤✉♥ tr➯♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ●å✐ M = M0 ⊃ M1 ⊃ ⊃ Mk = {0} ❧➔ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ M ✈ỵ✐ l(M ) = k ✈➔ M = M0 ⊃ M1 ⊃ ⊃ Mt = {0} ❧➔ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ M ✈ỵ✐ l(M ) = t✳ ❳➨t ❞➣②✿ M = g −1 (M ) ⊃ g −1 (M1 ) ⊃ ⊃ g −1 (Mt ) = f (M ) ⊃ f (M1 ) ⊃ ⊃ f (Mk ) = {0} ❧➔ ♠ët ♠ỉ✤✉♥ ✤ì♥ ✈ỵ✐ ♠é✐ ❉➵ t❤➜② g−1 Mi g−1(Mi+1) ∼ = Mi M i+1 i✳ ❉♦ ✤â ❞➣② tr➯♥ ❧➔ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ M ✈ỵ✐ ✤ë ❞➔✐ k + t✱ ♥❣❤➽❛ l(M ) = l(M ) + l(M )✳ ✷✳✸✳✶✳✷ ❍➺ q✉↔ ♠ët R ◆➳✉ R ✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ R ✲ ♠æ✤✉♥ M t❤➻ M ❧➔ l(M ) = l(M ) + l(M/M ) ✷✳✸✳✷ ✣✐➲✉ ❦✐➺♥ ✤➸ ♠æ✤✉♥ ❝â ỳ ỵ sỷ B ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠ët R ✲ ♠æ✤✉♥ A✳ ❑❤✐ ✤â R A ❧➔ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ B ✈➔ A/ B ❧➔ ♥❤ú♥❣ ✲ ♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ❍ì♥ ♥ú❛ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â l(A) = l(B) + l(A/B )✳ ✸✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❈❤ù♥❣ ♠✐♥❤✳ ❑❤✐ B = ❤♦➦❝ B = A ✤✐➲✉ ♥➔② ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❇➙② ❣✐í t❛ ❣✐↔ sû A ❧➔ ♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ A ⊃ B ⊃ ❧➔ ởt ổ A ỵ ❏♦r❞❛♥ ✲ ❤♦❧❞❡r ❝â t❤➸ ❜ê s✉♥❣ ✤➸ ❞➣② tr➯♥ trð t❤➔♥❤ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ A ⇒ A = M0 ⊃ M1 ⊃ ⊃ B = Mk ⊃ Mk+1 ⊃ ⊃ Mn = ❑❤✐ ✤â rã r➔♥❣ tø t❤➔♥❤ ♣❤➛♥ B = Mk ✤➳♥ ❝✉è✐ ❞➣② tr➯♥ ❧➔ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ B ✱ tù❝ B ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ ❞➣② A/ = M0/ ⊃ M1/ ⊃ ⊃ Mk/ = ✭✶✮ B B B B ❧➔ ♠ët ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ A/B ✳ ✣✐➲✉ ♥➔② ✤÷đ❝ s r tứ ỵ ổ t ✈➟② ❚❛ ❝â Mi/B Mi+1/B ∼ = Mi/Mi+1 , ∀i = 0, 1, , k − ❉♦ Mi/Mi+1 ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➯♥ Mi/B Mi+1/B ❝ơ♥❣ ❧➔ ❘ ✲ ♠ỉ✤✉♥ ✤ì♥✳ ◆❤÷ ✈➟② ✭✶✮ ❧➔ ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ A/B ✳ ❚ø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ❝ô♥❣ s✉② r❛ A/B ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ l(A) = l(B) + l(A/B )✳ ⇐ ●✐↔ sû B = M0 ⊃ M1 ⊃ ⊃ Mk = ✈➔ A/B = N0 ⊃ N1 ⊃ ⊃ Nt = ❧➔ ❤❛✐ ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ ❇ ✈➔ A/B ✳ ●å✐ π : A → A/B ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝❤➼♥❤ t➢❝✳ ❉♦ ✤â π ❧➔ t♦➔♥ ❝➜✉✳ ❱➔ ✤➦t Ni = π−1(Ni ), i = 0, 1, , t − ❑❤✐ ✤â t❛ ❝â✿ π(Bi) = Bi ✈➔ A = N0 ⊃ N1 ⊃ ⊃ Nt−1 ⊃ N ❱➻ Ni/ Ni B Ni+1/ ∼ = Ni/Ni+1 t❛ s✉② Ni+1 ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➯♥ tø ✤➥♥❣ ❝➜✉ B ✸✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ r❛ Ni/Ni+1 ❧➔ ♥❤ú♥❣ ❘ ✲ ♠ỉ✤✉♥ ✤ì♥✳ ❱➟② ❞➣② A = N0 ⊃ N1 ⊃ ⊃ Nt−1 ⊃ B = M0 ⊃ M1 ⊃ ⊃ Mk = ❧➔ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤ ❝õ❛ ❆✱ tù❝ ❧➔ ❆ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ l(A) = l(B) + l(A/B ) ỵ R f0 f1 fn−1 f2 fn → M1 → M2 → → Mn → ❧➔ ♠ët ❞➣② ❦❤ỵ♣ ✲ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ❑❤✐ ✤â✿ n (−1)i l(Mi ) = i=1 ❈❤ù♥❣ ♠✐♥❤✳ ✳ f ❚ø ❣✐↔ t❤✐➳t t❛ s✉② r❛ ❞➣② Mi−1 f→ Mi → Mi+1 ❧➔ ❞➣② ❦❤ỵ♣✱ ✈ỵ✐ i = 1, n ●å✐ ji : Imfi−1 → Mi ❧➔ ✤ì♥ ❝➜✉ ❝❤➼♥❤ t➢❝✳ ❑❤✐ ✤â t❛ ❝â ❞➣② ❦❤ỵ♣ j f ♥❣➢♥ → Imfi−1 → Mi → Imfi → ✭ ✈➻ Kerfi = Imfi−1 = Imji ✮ ⑩♣ ❞ö♥❣ ✤à♥❤ ỵ t õ l(Mi ) = l(Imfi1 ) + l(Imfi )✱ ✈ỵ✐ ♠å✐ i = 1, n ▲➜② tê♥❣ ✤❛♥ ❞➜✉ ❝→❝ ✤➥♥❣ t❤ù❝ i−1 i i i l(M1 ) = l(Imf0 ) + l(Imf1 ) l(M2 ) = l(Imf1 ) + l(Imf2 ) ✳✳✳ l(Mn ) = l(Imfn−1 ) + l(Imfn ) ❚❛ t❤✉ ✤÷đ❝ n i=1 (−1)i l(Mi ) = −l(Imf0 ) + (−1)n l(Imfn ) = 0✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✸✳✸ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ỵ ởt R ổ M õ ❞➔✐ ❤ú✉ ❤↕♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M ✈ø❛ ❧➔ ♠æ✤✉♥ ◆♦❡t❤❡r✱ ✈ø❛ ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ✸✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ M ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ♥➯♥ t ỵ r r t t ❤♦➦❝ ❣✐↔♠ t❤ü❝ sü tr♦♥❣ M ✤➲✉ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ❉♦ ✤â ♠å✐ ❞➣② t➠♥❣ ❤♦➦❝ ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✤➲✉ ♣❤↔✐ ❞ø♥❣✳ ❙✉② r❛ M ✈ø❛ ❧➔ ♠æ✤✉♥ ◆♦❡t❤❡r✱ ✈ø❛ ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ⇐ ●✐↔ sû M ✈ø❛ ❧➔ ♠æ✤✉♥ ◆♦❡t❤❡r✱ ✈ø❛ ❧➔ ♠æ✤✉♥ ❆rt✐♥✳ ❑❤✐ ✤â ♥➳✉ M ❧➔ ♠æ✤✉♥ ❦❤æ♥❣ t❤➻ M ❝â ✤ë ❞➔✐ ❜➡♥❣ 0✳ ●✐↔ sû M ❦❤→❝ ♠æ✤✉♥ ❦❤æ♥❣✳ M ổ tr tỗ t ởt ổ ỹ M1 M rỗ tỗ t ♠ët ♠æ✤✉♥ ❝♦♥ ❝ü❝ ✤↕✐ M2 ❝õ❛ M1 ✱✳✳✳ ❚ø ✤â t❛ ✤÷đ❝ ♠ët ❞➣② t❤ü❝ sü ❣✐↔♠ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ⇒ M = M0 ⊃ M1 ⊃ ⊃ Md ⊃ ▼➦t ❦❤→❝ M ❧➔ ♠æ✤✉♥ ❆rt✐♥✱ ♥➯♥ ❞➣② ❦❤æ♥❣ t❤➸ ❞➔✐ ✈æ ❤↕♥✳ ❱➻ ✈➟②✱ t❛ t❤✉ ✤÷đ❝ ♠ët ❞➣② ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✿ M = M0 ⊃ M1 ⊃ ⊃ Mn = {0}✭✯✮ tr♦♥❣ ✤â Mi ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝ü❝ ✤↕✐ ❝õ❛ Mi−1✳ ❚ù❝ ❧➔ Mi−1/Mi ❧➔ ♠ỉ✤✉♥ ✤ì♥✱ ✈ỵ✐ ♠å✐ i = 1, n ❉♦ ✤â ✭✯✮ ❧➔ ♠ët ❞➣② ❤ñ♣ t❤➔♥❤✳ ❙✉② r❛ M ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ✸✾ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q s ữ r ởt tố ỵ tt ỵ t t ♠➺♥❤ ✤➲ ❧➔♠ ❝ì sð ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ✲ ❚ø ❦❤→✐ ♥✐➺♠ ❞➣② ❤đ♣ t❤➔♥❤ ❝õ❛ ♠ỉ✤✉♥ ✤÷❛ r❛ ✤à♥❤ ♥❣❤➽❛ ✈➲ ✤ë ❞➔✐ ♠ỉ✤✉♥✳ ✲ ✣÷❛ r❛ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ♥❤÷✿ t➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ ✤ë ❞➔✐✱ ✤✐➲✉ ❦✐➺♥ ✤➸ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❤↕♥ ❝❤➳ ♥➯♥ ♥❤✐➲✉ ✈➜♥ ✤➲ ✈➲ ✤ë ❞➔✐ ♠ỉ✤✉♥ ❝❤÷❛ ữủ q t ữ ợ ổ ✤➦❝ ❜✐➺t✿ ♠æ✤✉♥ ♥ë✐ ①↕✱ ♠æ✤✉♥ ①↕ ↔♥❤✱ ♠æ✤✉♥ tü ❞♦✳✳✳ ❤♦➦❝ ♠è✐ ❧✐➯♥ ❤➺ ❝õ❛ ✤ë ❞➔✐ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t tr♦♥❣ ♠æ✤✉♥✳ ❊♠ ❤② ✈å♥❣ ✤➙② ❧➔ ❝→❝ ✈➜♥ ✤➲ ✤➸ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ②➯✉ t❤➼❝❤ ♠æ♥ ✣↕✐ sè t✐➳♣ tư❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ❡♠ ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ỳ t sõt rt qỵ ổ ũ t t õ õ ỵ õ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥ ♥ú❛✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ✹✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●❱❍❉✿ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ [1] ◆❣✉②➵♥ ❚ü ❈÷í♥❣✱ ●✐→♦ tr➻♥❤ ✣↕✐ sè ❤✐➺♥ ✤↕✐✱ ◆❳❇ ✣↕✐ ❍å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✱ ✷✵✵✸✳ [2] ỗ r P tr số ❣✐❛♦ ❤♦→♥✱ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱ ◆❳❇ ✣↕✐ ❤å❝ ❱✐♥❤✱ ✷✵✵✼✳ [3] ❘✳❨✳❙❤❛r♣✱ ❙t❡♣s ✐♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ s❡❝♦♥❞ ❡❞✐t✐♦♥✱ ❈❛♠❜✳ Prss [4] ữỡ ố t ỵ tt ổ ◆❳❇ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐✱ ✷✵✵✽✳ ✹✶ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ====== NGUYỄN THỊ KIM ANH VỀ MƠĐUN CĨ ĐỘ DÀI HỮU HẠN TRÊN VÀNH GIAO HỐN KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Người hướng dẫn khoa