(LUẬN văn THẠC sĩ) tính catenary trong các đại số hữu hạn sinh như môđun

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM VĂN CHUYỀN TÍNH CATENARY TRONG CÁC ĐẠI SỐ HỮU HẠN SINH NHƯ MƠĐUN LUẬN VĂN THẠC SĨ TỐN HỌC THÁI NGUYÊN - 2015 download by : skknchat@gmail.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM VĂN CHUYỀN TÍNH CATENARY TRONG CÁC ĐẠI SỐ HỮU HẠN SINH NHƯ MÔĐUN Chuyên ngành: Đại số Lý thuyết số Mã số: 62.46.01.04 LUẬN VĂN THẠC SĨ TOÁN HỌC Người hướng dẫn khoa học: PGS.TS NGUYỄN THỊ DUNG THÁI NGUYÊN - 2015 download by : skknchat@gmail.com ▲ê✐ ❝❛♠ ➤♦❛♥ ✐ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❦❤➠♥❣ ❜Þ trï♥❣ ❧➷♣ ✈í✐ ❝➳❝ ❧✉❐♥ ✈➝♥ tr➢í❝ ➤➞②✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ❧➭ ❝➳❝ ♥❣✉å♥ t➭✐ ❧✐Ö✉ ♠ë✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t➭✐ ❧✐Ö✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❣❤✐ râ ♥❣✉å♥ ❣è❝✳ ❍ä❝ ✈✐➟♥ P❤➵♠ ❱➝♥ ❈❤✉②Ị♥ download by : skknchat@gmail.com ▲ê✐ ❝➯♠ ➡♥ ✐✐ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤ ✈➭ s➞✉ s➽❝ tớ P ễ ị t tì ❤➢í♥❣ ❞➱♥ t➠✐ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ tr♦♥❣ ❱✐Ö♥ ❚♦➳♥ ❤ä❝ ❱✐Öt ◆❛♠✱ tr♦♥❣ ❦❤♦❛ ❚♦➳♥ ✈➭ ❜é ♣❤❐♥ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❚❤➳✐ ◆❣✉②➟♥✱ ➤➲ t❐♥ t×♥❤ ❣✐➯♥❣ ❞➵②✱ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❳✐♥ ❝➯♠ ➡♥ ♥❤÷♥❣ ♥❣➢ê✐ t❤➞♥ tr♦♥❣ ❣✐❛ ➤×♥❤✱ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✷✶✱ ➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ❇➯♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ĩ tr➳♥❤ ỏ ữ tế sót rt ợ ỉ ❜➯♦ t❐♥ t×♥❤ ❝đ❛ ❝➳❝ t❤➬② ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ö♣✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✺ ♥➝♠ ✷✵✶✺ ❚➳❝ ❣✐➯ ❧✉❐♥ ✈➝♥ P❤➵♠ ❱➝♥ ❈❤✉②Ò♥ download by : skknchat@gmail.com ✐✐✐ ▼ô❝ ❧ô❝ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ✶✳ ➜➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥ ✶ ✳✳✳✳✳ ✸ ✶✳✶✳ P❤ô t❤✉é❝ ♥❣✉②➟♥ tr➟♥ ✈➭♥❤ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ➜➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét ❈❤➢➡♥❣ ✷✳ R✲♠➠➤✉♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶✸ ❚Ý♥❤ ❝❛t❡♥❛r② tr♦♥❣ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✷✶ ✷✳✶✳ ❚Ý♥❤ ❝❛t❡♥❛r② ✈➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳ ❚Ý♥❤ ❝❛t❡♥❛r② ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ download by : skknchat@gmail.com ✶ ▼ë ➤➬✉ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r ✈➭ p ⊂ q ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R ▼ét ❞➲② t➝♥❣ ❝❤➷t ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè p = p0 ⊂ p1 ⊂ ⊂ pn = q ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ①Ý❝❤ tố ò ữ tồ t ột ♥❣✉②➟♥ tè p ✈➭ q ♥Õ✉ ✈í✐ ♠ä✐ i✱ ❦❤➠♥❣ p s❛♦ ❝❤♦ pi ⊂ p ⊂ pi+1 ❑❤✐ ➤ã n ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ❞➭✐ ❝đ❛ ①Ý❝❤ ✐➤➟❛♥ tố ò tr R ợ ọ ✈➭♥❤ ❝❛t❡♥❛r② ♥Õ✉ ✈í✐ ♠ä✐ ❝➷♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè p R ❧✉➠♥ tå♥ t➵✐ ①Ý❝❤ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❜➲♦ ❤ß❛ ⊂ q ❝ñ❛ p = p0 ⊂ p1 ⊂ ⊂ pn = q ♥➺♠ ❣✐÷❛ p ✈➭ q ✈➭ ♠ä✐ ①Ý❝❤ ♥❣✉②➟♥ tè ❜➲♦ ❤♦➭ ❣✐÷❛ ❣ä✐ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥Õ✉ ♠ä✐ p ✈➭ q ➤Ị✉ ó ộ R ợ R số ữ ❤➵♥ s✐♥❤ ➤Ị✉ ❧➭ ❝❛t❡♥❛r②✳ ❚Ý♥❤ ❝❛t❡♥❛r② ❝đ❛ ❝➳❝ ✈➭♥❤ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ➤➬✉ t✐➟♥ ❜ë✐ ❲✳ ❑r✉❧❧ ❬❑r❪ ♥➝♠ ✶✾✸✼✳ ❙❛✉ ➤ã ❤➭♥❣ ❧♦➵t ❝➠♥❣ tr×♥❤ ❝đ❛ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ♥❤➢ ▼✳ ◆❛❣❛t❛ ❬◆✶❪✱ ❬◆✷❪✱ ■✳ ❙✳ ❈♦❤❡♥ ❬❈✶❪✱ ❬❈✷❪✱ ▲✳ ❏✳ ❘❛t❧✐❢❢ ❬❘✶❪✱ ❬❘✷❪✱ ✳✳✳ ❝❤♦ t❤✃② r➺♥❣ tÝ♥❤ ❝❤✃t ♥➭② ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ✈➭♥❤ ✈➭ ♠➠➤✉♥✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ❲✳ ❑r✉❧❧ ❬❑r❪ ✭✶✾✸✼✮ ➤➲ ❝❤ø♥❣ tá r➺♥❣ ♠ä✐ ➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ♠ét tr➢ê♥❣ ❧➭ ❝❛t❡♥❛r②✳ ❚✐Õ♣ t❤❡♦✱ ■✳ ❙✳ ❈♦❤❡♥ ❬❈✶❪ ✭✶✾✹✻✮ ❝❤ø♥❣ ♠✐♥❤ ♠ä✐ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➬② ➤đ ➤Ị✉ ❧➭ ❝❛t❡♥❛r②✳ ❑Õt q✉➯ t✐Õ♣ t❤❡♦ t❤✉é❝ ✈Ò ▼✳ ◆❛❣❛t❛ ❬◆✶❪ ✭✶✾✺✻✮ ♥ã✐ r➺♥❣ ♠ä✐ ♠✐Ị♥ ♥❣✉②➟♥ ➤Þ❛ ♣❤➢➡♥❣ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ❧➭ ❝❛t❡♥❛r②✳ ◆❣♦➭✐ r❛✱ ❜➺♥❣ ♥❤÷♥❣ ❧❐♣ ❧✉❐♥ ➤➡♥ ❣✐➯♥✱ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❧í♣ ✈➭♥❤ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ R/I R s❛♦ ❝❤♦ dim R ❧➭ ✈➭♥❤ ❝❛t❡♥❛r②✱ ❧➭ ❝❛t❡♥❛r② ✈í✐ ♠ä✐ ✐➤➟❛♥ I ❝đ❛ R ❧➭ ❝❛t❡♥❛r② R✱ ➯♥❤ ➤å♥❣ ❝✃✉ ✈➭ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ❝đ❛ ✈➭♥❤ ❝❛t❡♥❛r② ❝ị♥❣ ❧➭ ❝➳❝ ✈➭♥❤ ❝❛t❡♥❛r②✳ ◆❤➢ ✈❐② ❤➬✉ ❤Õt ❝➳❝ ✈➭♥❤ ➤➢ỵ❝ ❜✐Õt ➤Õ♥ tr♦♥❣ t❤ù❝ tế tr ữ ứ ụ ủ ì ọ sè ➤Ị✉ ❧➭ ❝❛t❡♥❛r②✳ ❱× ✈❐②✱ ♠ét ❝➞✉ ❤á✐ ➤➷t r❛ ❧➭ ❧✐Ư✉ r➺♥❣ tÝ♥❤ ❝❛t❡♥❛r② ❝ã ➤➢ỵ❝ ❜➯♦ t♦➭♥ q✉❛ ❝➳❝ ♠ë ré♥❣ ✈➭♥❤❄ ❱Ý ❞ơ ♥Õ✉ ❝❤♦ ❤÷✉ ❤➵♥ s✐♥❤ t❤× ❧✐Ư✉ r➺♥❣ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ Λ ❧➭ ♠ét R✲♠➠➤✉♥ R ✈➭ Λ ❧✐➟♥ ❤Ư ✈í✐ ♥❤❛✉ ♥❤➢ t❤Õ download by : skknchat@gmail.com ✷ ♥➭♦❄ ◆➝♠ ✶✾✾✾✱ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪ tr♦♥❣ ❜➭✐ ❜➳♦ ✧❈❆❚❊◆❆❘■❚❨ ■◆ ▼❖❉❯▲❊✲❋■◆■❚❊ ❆▲●❊❇❘❆❙✧ ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ➤➲ tr➯ ❧ê✐ ❝❤♦ ❝➞✉ ❤á✐ tr➟♥✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❜➭✐ ❜➳♦ ❧➭ ➤Þ♥❤ ❧ý s❛✉✳ ➜Þ♥❤ ❧ý✳ ●✐➯ sö R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✱ Λ ❧➭ R✲➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥ ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ❧➭ ❣✐❛♦ ❤♦➳♥✮✳ ❑❤✐ ➤ã Λ ❧➭ ❝❛t❡♥❛r② ♥Õ✉ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❚❤❡♦ ❘❛t❧✐❢❢ ❬❘✷✱ ➜Þ♥❤ ý ế R ề ị tì tí ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ tÝ♥❤ tù❛ ❦❤➠♥❣ tré♥ ì tế ề ợ ủ ị ý tr ❝ị♥❣ ➤ó♥❣ ♥Õ✉ R ❧➭ ♠✐Ị♥ ➤Þ❛ ♣❤➢➡♥❣ ✭❍Ư q✉➯ ✷✳✷✳✾✮✳ ❱× ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥➟♥ tõ ➤Þ♥❤ ❧ý tr➟♥ t❛ ❝ã ♥❣❛② ♠ét ❤Ư q✉➯ ❧➭ ế R tì ữ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❧➭ Λ ❧➭ ❝❛t❡♥❛r② ✭❍Ư q✉➯ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈➭ Λ R✲♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× Λ ❧➭ ❝❛t❡♥❛r② ✭❍Ư q✉➯ ✷✳✷✳✽✮✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ tr×♥❤ ❜➭② ❧➵✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ❦Õt q✉➯ ✈Ị R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ tÝ♥❤ ❝❤✃t ❝❛t❡♥❛r②✱ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ✈➭♥❤ ✈➭ ❦Õt q✉➯ ❝đ❛ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪ ✈Ị tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ❞➭♥❤ ➤Ĩ tr×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ tÝ♥❤ ❝❤✃t ❧✐➟♥ q✉❛♥ ➤Õ♥ ♠ë ré♥❣ ♥❣✉②➟♥ tr➟♥ ♠ét ị ý s s ợ ị ý ♥➺♠ tr➟♥✱ ➤Þ♥❤ ❧ý ❣♦✐♥❣✲✉♣✱ ✳✳✳ ❈➳❝ ❦❤➳✐ ♥✐Ư♠✱ ✈Ý ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s ũ ợ trì tr ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ò tÝ♥❤ ❝❛t❡♥❛r②✱ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ❤❛✐ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❜➭✐ ❜➳♦ ❝ñ❛ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪✳ P❤➬♥ ❦Õt ❧✉❐♥ ❝đ❛ ❧✉❐♥ ✈➝♥ tỉ♥❣ ❦Õt ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➵t ➤➢ỵ❝✳ download by : skknchat@gmail.com ✸ ❈❤➢➡♥❣ ✶ ➜➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R ✲♠➠➤✉♥ ◆é✐ ❞✉♥❣ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② ❣å♠ ❤❛✐ ♣❤➬♥✱ ♣❤➬♥ ♠ét ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ✈Ị ♠ë ré♥❣ ♥❣✉②➟♥ ✈➭ ❤Ư t❤è♥❣ ❧➵✐ ♠ét sè ❦Õt q✉➯ ❝➡ së ❝ñ❛ ♠ë ré♥❣ ị ý s s ợ rt ị ❧ý ♥➺♠ tr➟♥ ✭▲②✐♥❣✲ ♦✈❡r✮✱ ➜Þ♥❤ ❧ý ➤✐ ❧➟♥ ✭●♦✐♥❣✲✉♣✮✳✳✳ P❤➬♥ ❤❛✐ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠✱ ✈Ý ❞ơ ✈➭ ❤Ư t❤è♥❣ ❧➵✐ ♠ét sè ❦Õt q✉➯ ❝❤Ý♥❤ ✈Ò ❧✉➠♥ ❦Ý ❤✐Ư✉ R✱ S R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❧➭ ❝➳❝ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ M ❧➭ ♠ét R✲♠➠➤✉♥ ✈➭ Λ ❧➭ ♠ét R✲➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R ết q tt ữ ợ t tr ❝✉è♥ s➳❝❤ ❝ñ❛ ❘✳ ❨✳ ❙❤❛r♣ ❬❙❤❪✱ ▼✳ ❋✳ ❆t✐②❛❤ ❛♥❞ ■✳ ●✳ ▼❛❝❉♦♥❛❧❞ ❬❆▼❪ ✈➭ ❜➭✐ ❜➳♦ ❝ñ❛ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪✳ ✶✳✶✳ P❤ô t❤✉é❝ ♥❣✉②➟♥ tr➟♥ ✈➭♥❤ ❝♦♥ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳ ❤♦➳♥ S✱ ♣❤➬♥ tư r0 , , rh−1 ∈ R ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❱➭♥❤ tr➟♥ S ✭❬❙❤✱ ➜Þ♥❤ ♥❣❤Ü❛ ✶✸✳✶✻❪✮ ❈❤♦ s ∈ S R ❧➭ ✈➭♥❤ ❝♦♥ ❝ñ❛ ✈➭♥❤ ❣✐❛♦ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥❣✉②➟♥ tr➟♥ s❛♦ ❝❤♦ sh R ♥Õ✉ tå♥ t➵✐ h ∈ N ✈➭ + rh−1 sh−1 + + r0 = 0, ❤❛② ♥ã✐ ❝➳❝❤ ❦❤➳❝ s ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♠ét ➤❛ t❤ø❝ ♠♦♥✐❝ tr♦♥❣ ✈➭♥❤ R[X] ➤➢ỵ❝ ❣ä✐ ❧➭ ♥❣✉②➟♥ tr➟♥ R ♥Õ✉ ♠ä✐ ♣❤➬♥ tư s ∈ S ➤Ị✉ ❧➭ ♥❣✉②➟♥ R✳ ▼ét f :RR ữ ợ ❣ä✐ ❧➭ ♥❣✉②➟♥ download by : skknchat@gmail.com ✹ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ R ❧➭ ♥❣✉②➟♥ tr➟♥ ✈➭♥❤ ❝♦♥ ◆❤➽❝ ❧➵✐ r➺♥❣ ♠ét R✲♠➠➤✉♥ M Im(f ) ❝đ❛ ♥ã✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ❢❛✐t❤❢✉❧ ♥Õ✉ (0 :R M ) = 0✳ ❑❤✐ ➤ã t❛ ❝ã ❦Õt q✉➯ s❛✉✿ ▼Ư♥❤ ➤Ị ✶✳✶✳✷✳ ❤♦➳♥ ✭✐✮ ✭❬❙❤✱ ▼Ư♥❤ ➤Ị ✶✸✳✷✵❪✮ ❈❤♦ R ❧➭ ♠ét ✈➭♥❤ ❝♦♥ ❝ñ❛ ✈➭♥❤ ❣✐❛♦ S ✈➭ s ∈ S ✳ ữ ị s t s ♥❣✉②➟♥ tr➟♥ R❀ ✭✐✐✮ ❱➭♥❤ ❝♦♥ R[s] ❝ñ❛ S ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥❀ ✭✐✐✐✮ ❚å♥ t➵✐ ✈➭♥❤ ❝♦♥ T ❝đ❛ S ❝❤ø❛ R[s] ✈➭ T ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥❀ ✭✐✈✮ ❚å♥ t➵✐ ♠ét R[s]✲♠➠➤✉♥ ❢❛✐t❤❢✉❧ M ✱ ♠➭ ♥Õ✉ ❝♦✐ M ♥❤➢ R✲♠➠➤✉♥ ❜ë✐ é t ẹ tì M ữ s✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ♥❤➢ (i) ⇒ (ii)✳ R✲♠➠➤✉♥✳ ❚❤❐t ✈❐②✱ ✈× s R[s] ❧➭ ♥❣✉②➟♥ tr➟♥ R s✐♥❤ ❜ë✐ {si : i ∈ N0 } ♥➟♥ tå♥ t➵✐ h ∈ N ✈➭ r0 , , rh−1 ∈ R t❤á❛ ♠➲♥ sh + rh−1 sh−1 + + r0 = ❉♦ ➤ã ✈í✐ ♠ä✐ n ∈ N0 t❛ ❝ã sh+n = −rh−1 sh+n−1 − − r0 sn ❇➺♥❣ q✉② ♥➵♣ t❤❡♦ n t❛ ❝ã t❤Ĩ ❞Ơ ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ sh+n ∈ R1 + Rs + + Rsh−1 ➤✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭ ♠ä✐ ♣❤➬♥ tư ❝đ❛ R[s] ➤Ị✉ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ q✉❛ 1, s, , sh−1 ✱ ❝❤ø♥❣ tá R[s] ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ R✲♠➠➤✉♥✳ (ii) ⇒ (iii)✳ ❚❛ ❝❤Ø ❝➬♥ ❝❤ä♥ T = R[s]✳ download by : skknchat@gmail.com ✺ (iii) ⇒ (iv) ▲✃② M = T ✳ ❑❤✐ ➤ã T ❣✐➯ t❤✐Õt t❤❡♦ (iii)✳ ❧➭ ♠ét ▼➷t ❦❤➳❝✱ ❧✃② ❜✃t ❦× ♣❤➬♥ tư a1R = s✉② r❛ a = ❱× t❤Õ✱ M = T (iv) ⇒ (i) ❈❤♦ M ♠➠➤✉♥ s✐♥❤ ❜ë✐ tå♥ t➵✐ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ t❤❡♦ a ∈ (0 :R[s] T ), ❧➭ tõ aT = ❦Ð♦ R[s]✲❢❛✐t❤❢✉❧✳ ❧➭ ♠ét R[s]✲♠➠➤✉♥ ❢❛✐t❤❢✉❧ ✈➭ M ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ R✲ u1 , u2 , , un ✳ ❑❤✐ ➤ã ✈× sui ∈ M ỗ i = 1, , n✱ aij ∈ R t❤á❛ ♠➲♥ n suj = ✭✶✮ aij ui ➜➷t ♠❛ tr❐♥ ❝ñ❛ A✱ A = (aij )✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ s ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ➤❛ t❤ø❝ ➤➷❝ tr➢♥❣ tø❝ ❧➭ s ❧➭ ♥❣❤✐Ư♠ ❝đ❛ f (t) = det(tIn − A) ❚❤❐t ✈❐②✱ tõ ➤➻♥❣ t❤ø❝ ✭✶✮ t❛ ❝ã t❤Ĩ ✈✐Õt ❞➢í✐ ❞➵♥❣ (u1 , , un )sIn = (u1 , , un )A ❤❛② ✭✷✮ (u1 , , un )(sIn − A) = (0, , 0) ❑Ý ❤✐Ö✉ (sIn − A)ad ❧➭ ♠❛ tr❐♥ ❧✐➟♥ ❤ỵ♣ ❝đ❛ ♠❛ tr❐♥ (sIn − A)✳ ❚❤❡♦ ❬❙❤✱ ❍Ö q✉➯ ✶✸✳✶✷❪✱ t❛ ❝ã tÝ♥❤ ❝❤✃t (sIn − A)(sIn − A)ad = det(sIn − A)In ◆❤➞♥ ❝➯ ❤❛✐ ✈Õ ❝đ❛ ✭✷✮ ✈í✐ (sIn − A)ad rå✐ ➳♣ ❞ơ♥❣ tÝ♥❤ ❝❤✃t tr➟♥ t❛ ➤➢ỵ❝ (u1 , , un )(sIn − A)(sIn − A)ad = (0, , 0)(sIn − A)ad ❙✉② r❛ (u1 , , un ) det(sIn − A)In = (0, , 0) ❤❛② (u1 , , un )f (s)In = (f (s)u1 , , f (s)un )In = (0, , 0) download by : skknchat@gmail.com ✷✹ ✈í✐ ♠ä✐ 0≤i ht p + ❚❤❐t ✈❐②✱ ➤➷t ht m/P = d✳ ❳Ðt tr♦♥❣ ✈➭♥❤ R/p✱ t❛ ❝ã t❤Ó ❝❤Ø r❛ r➺♥❣ tå♥ t➵✐ ✈➠ sè Pλ ∈ Spec R t❤á❛ ♠➲♥✿ p ⊂ Pλ ✱ ht(Pλ /p) = ✈➭ ht(m/Pλ ) = d✳ download by : skknchat@gmail.com ❍➡♥ t❤Õ ✷✺ ♥÷❛ t❤❡♦ ❣✐➯ t❤✐Õt✱ ht(m/Pλ ) + ht Pλ = n✱ ❞♦ ➤ã ht Pλ = n − d = ht P > ht p + ▼➷t t ị ý ỉ ó ữ ❝➳❝ ✐➤➟❛♥ ➤ã t❛ ❝ã ♠➞✉ t❤✉➱♥✳ ❱❐② Pλ ♥❤➢ t❤Õ✳ ❉♦ R ❧➭ ❝❛t❡♥❛r②✳ ❍➡♥ ♥÷❛✱ ✈➭♦ ♥➝♠ ✶✾✼✼ ❦Õt q✉➯ tr➟♥ ➤➲ ➤➢ỵ❝ ▼❝❆❞❛♠ ✈➭ ❘✳ ❏✳ ❘❛t❧✐❢❢ ❬▼❝❘❪ ♠ë ré♥❣ ❝❤♦ t✃t ❝➯ ❝➳❝ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➻♥❣ ❝❤✐Ị✉✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ✈➭♥❤ R ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➻♥❣ ❝❤✐Ị✉ ♥Õ✉ dim R/p = dim R ✈í✐ ♠ä✐ p ∈ Min Ass R ➜Þ♥❤ ❧ý ✷✳✶✳✻✳ ➤ã ❬▼❝❘❪ ●✐➯ sư R ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ➤➻♥❣ ❝❤✐Ị✉✳ R tr ỉ ỗ ✐➤➟❛♥ ♥❣✉②➟♥ tè p ❝ñ❛ R t❛ ❝ã ht p + dim R/p = dim R ❚✐Õ♣ t❤❡♦ ❝❤ó♥❣ t❛ t×♠ ❤✐Ĩ✉ ✈Ị ♠ét ❧♦➵✐ ➤➷❝ ❜✐Ưt ❝đ❛ ❧í♣ ✈➭♥❤ tr tr ổ ụ ợ ị ĩ s ị ĩ R ợ ọ tr ♣❤ỉ ❞ơ♥❣ ♥Õ✉ ♠ä✐ R✲➤➵✐ sè ❱➭♥❤ ❤÷✉ ❤➵♥ s✐♥❤ ➤Ị✉ ❧➭ ❝❛t❡♥❛r②✳ ▼Ư♥❤ ➤Ị ✷✳✶✳✽✳ ✭✐✮ ❱➭♥❤ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♠ä✐ ✈➭♥❤ ➤❛ t❤ø❝ ❤÷✉ ❤➵♥ ❜✐Õ♥ ✈í✐ ❤Ư sè tr➟♥ R ❧➭ ❝❛t❡♥❛r②✳ ✭✐✐✮ ❱➭♥❤ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ R/p ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈í✐ ♠ä✐ p ∈ Spec R✳ ✭✐✐✐✮ ➯♥❤ ➤å♥❣ ❝✃✉ ✈➭ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ❝đ❛ ✈➭♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝ị♥❣ ❧➭ ✈➭♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ●✐➯ sư a1 , , an ∈ S ✈➭♥❤ s❛♦ ❝❤♦ S ❧➭ ♠ét R✲➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤✱ tø❝ ❧➭ tå♥ t➵✐ S = R[a1 , , an ]✳ f : R[X1 , , Xn ] → R[a1 , , an ] ✈í✐ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét t♦➭♥ ❝✃✉ f (Xi ) = , i = n✱ download by : skknchat@gmail.com ✈× t❤Õ ✷✻ S ➤➻♥❣ ❝✃✉ ✈í✐ ♠ét ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ➤❛ t❤ø❝ R[X1 , , Xn ]✳ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❝❛t❡♥❛r② ❧➭ ❝❛t❡♥❛r② ♥➟♥ t❛ s✉② r❛ r➺♥❣ ✈➭♥❤ R ❱× ✈➭♥❤ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♠ä✐ ✈➭♥❤ ➤❛ t❤ø❝ ❤÷✉ ❤➵♥ ❜✐Õ♥ ✈í✐ ❤Ư sè tr➟♥ R ❧➭ ❝❛t❡♥❛r②✳ ✭✐✐✮ ✈➭ ✭✐✐✐✮ ❞Ô ❞➭♥❣ s✉② r❛ tõ ✭✐✮ ✈➭ tÝ♥❤ ❝❤✃t ❝ñ❛ ✈➭♥❤ ❝❛t❡♥❛r②✳ ❙❛✉ ➤➞② ❧➭ ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ✈➭♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❚r➢í❝ ❤Õt✱ ❝❤ó♥❣ t❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ✈➭♥❤ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✈➭ ✈➭♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ t❤❡♦ t❤✉❐t ♥❣÷ ❝đ❛ ▼✳ ◆❛❣❛t❛ ❬◆✷❪✳ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✾✳ ✐✮ ❱➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ♥Õ✉ ✈➭♥❤ ➤➬② ➤đ t❤❡♦ ✈í✐ ♠ä✐ (R, m) ➤➢ỵ❝ ❣ä✐ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ m✲❛❞✐❝ R ❝đ❛ ♥ã ❧➭ ➤➻♥❣ ❝❤✐Ị✉✱ tø❝ dim R/p = dim R p ∈ Min Ass R✳ ✐✐✮ ❱➭♥❤ ❘ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ tré♥ ❧➱♥ ♥Õ✉ dim R/p = dim R ✈í✐ ♠ä✐ p ∈ Ass R✳ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✵✳ ❬▼❛✶✱ ➜Þ♥❤ ❧Ý ✸✶✳✻❪ ●✐➯ sö ✈➭♥❤ ✭R, m✮ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ❑❤✐ ➤ã ✭✐✮ Rp ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✈í✐ ♠ä✐ p ∈ Spec(R)❀ ✭✐✐✮ ◆Õ✉ I ❧➭ ✐➤➟❛♥ ❝ñ❛ R t❤× R/I ❧➭ ➤➻♥❣ ❝❤✐Ị✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ R/I ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥❀ ✭✐✐✐✮ R ❧➭ ❝❛t❡♥❛r② ♣❤æ ❞ô♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ❈❤♦ S ❧➭ ♣❤➻♥❣ tr➟♥ Rp P ∈ Spec R s❛♦ ❝❤♦ P ∩ R = p ✈➭ ➤➷t S = (R)P ✳ ♥➟♥ S ❧➭ ♣❤➻♥❣ tr➟♥ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Þ❛ ♣❤➢➡♥❣ ✈➭ S Rp ✱ ❞♦ ➤ã S ❱× ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ♠ét ➤➻♥❣ ❝❤✐Ị✉ t❤❡♦ ❬▼❛✶✱ ➜Þ♥❤ ❧ý ✶✺✳✺❪✳ ▼➷t ❦❤➳❝ t❛ ➤➲ ❜✐Õt ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ♠ét ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Þ❛ ♣❤➢➡♥❣ ➤➻♥❣ ❝❤✐Ị✉ t❤× ✈➭♥❤ ➤➬② ➤đ t❤❡♦ ❞♦ ➤ã S m✲❛❞✐❝ ❝đ❛ ♥ã ❝ị♥❣ ➤➻♥❣ ❝❤✐Ị✉ ✭❬▼❛✶✱ ❍Ư q✉➯ tr✳✷✺✶❪✮✱ ❝ị♥❣ ➤➻♥❣ ❝❤✐Ị✉ ❦Ð♦ t❤❡♦ Rp ❝ị♥❣ ➤➻♥❣ ❝❤✐Ị✉✳ ❱❐② Rp ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ download by : skknchat@gmail.com ✷✼ ✭✐✐✮ ❈❤♦ t➽❝ tõ I ❧➭ ✐➤➟❛♥ ❝ñ❛ R/I → R/I R ✈➭ R/I ❧➭ ➤➻♥❣ ❝❤✐Ò✉✳ ❳Ðt ➳♥❤ ①➵ ❝❤✐Õ✉ ❝❤Ý♥❤ ❧➭ ♠ét ➤å♥❣ ❝✃✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❱× ❧➭ ♣❤➻♥❣ tr➟♥ R/I R/I ♥➟♥ R/I ❧➭ ➤➻♥❣ ề tì R/I ề t ị ý ✸✶✳✺❪✮✳ ▼➷t ❦❤➳❝ ♥Õ✉ R/I ❧➭ ➤➻♥❣ ❝❤✐Ị✉ ✈× dim R/I = dim R/I ✈➭ p ❧➭ ✐➤➟❛♥ tè✐ t❤✐Ó✉ ❝đ❛ R/I t❤× p ❝ị♥❣ ❧➭ ✐➤➟❛♥ tè✐ t❤✐Ĩ✉ ❝đ❛ R/I ✈➭ dim(R/I)/p ❞♦ ➤ã R/I ❧➭ ➤➻♥❣ ❝❤✐Ò✉ ❤❛② ❝❤✐Ị✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✭✐✐✐✮ ●✐➯ sư ✈➭♥❤ R/I R/I = dim R/I/p✱ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ❱❐② R/I ❧➭ ➤➻♥❣ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ R ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ❚❤❡♦ ❬▼❛✶✱ ➜Þ♥❤ ❧Ý ✸✶✳✻✱ ✭✐✐✐✮❪✱ ọ ề ị ốt ế ể ữ tr➟♥ R ➤Ò✉ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✱ ❞♦ ➤ã ề tr ì ọ ề ữ s✐♥❤ tr➟♥ R ➤Ò✉ ❧➭ ❝❛t❡♥❛r②✱ ❤❛② R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❑Õt q✉➯ s❛✉ ❝❤♦ ❝❤ó♥❣ t❛ ➤✐Ị✉ ❦✐Ư♥ ➤Ĩ ♠ét ✈➭♥❤ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✶✳ ✭✐✮ ❬▼❛✶✱ ➜Þ♥❤ ❧Ý ✸✶✳✼❪ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣❀ ✭✐✐✮ ❱➭♥❤ ➤❛ t❤ø❝ ♠ét ❜✐Õ♥ ✭✐✐✐✮ R[x] ❧➭ ❝❛t❡♥❛r②❀ R/p ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✈í✐ ♠ä✐ p ∈ Spec(R)✳ ❈❤ø♥❣ ♠✐♥❤✳ (i) ⇒ (ii) ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ✈➭♥❤ ❝❛t❡♥r② ♣❤ỉ ❞ơ♥❣✳ (ii) ⇒ (iii) ❚❤❡♦ ❇ỉ ➤Ị ✸ ✈➭ ❇ỉ ➤Ị ✹ ❬▼❛✶✱ tr❛♥❣ ✷✺✷✲✷✺✹❪✳ (iii) ⇒ (i) ●✐➯ sư R/p ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✈í✐ ♠ä✐ p ❧➭ ✐➤➟❛♥ t❛ ➤➢ỵ❝ R ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ p ∈ Spec R✱ ❝❤ä♥ ➳♣ ❞ô♥❣ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✵ t❛ ❝ã R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❍Ư q✉➯ ✷✳✶✳✶✷✳ ◆Õ✉ dim R ≤ t❤× R ❧➭ ✈➭♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✹ t❛ ❝ã ♠ä✐ ✈➭♥❤ ❝ã ❝❤✐Ò✉ ♥❤á ❤➡♥ ❤♦➷❝ ❜➺♥❣ ✷ ❧➭ ❝❛t❡♥❛r②✱ ♥➟♥ ♥Õ✉ dim R ≤ t❤× dim R[x] ≤ 2✱ ❞♦ ➤ã R[x] ❧➭ ❝❛t❡♥❛r② download by : skknchat@gmail.com ✷✽ ✈➭ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✶ t❤× t❤× R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❱× ✈❐②✱ ♥Õ✉ dim R ≤ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ ❣✐❛♦ ❤♦➳♥✱ ♠ét R✲♠➠➤✉♥ M ❤➵♥ s✐♥❤ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥Õ✉ ❤÷✉ M = ❤♦➷❝ depth M = dim M ế R R tì R ợ ọ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ➤Þ❛ ♣❤➢➡♥❣✳ ❚❛ ❝ã ❝➳❝ ❦Õt q✉➯ s❛✉✳ ❇ỉ ➤Ị ✷✳✶✳✶✸✳ ✭✐✮ ❈❤♦ R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❦❤✐ ➤ã Rp ❝ò♥❣ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭✐✐✮ R[X1 , , Xn ] ❝ị♥❣ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✹✳ ❈❤♦ ✐➤➟❛♥ t❤ù❝ sù ❝ñ❛ ✭✐✮ (R, m) ❧➭ ♠ét ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤Þ❛ ♣❤➢➡♥❣✱ I ❧➭ R✳ ❑❤✐ ➤ã ht I = depth(I, R) ✈➭ ❞♦ ➤ã ht I + dim R/I = dim R✳ ✭✐✐✮ R ❧➭ ✈➭♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ ❝ã ▼❛❝❛✉❧❛② ♥➟♥ Rp ht I = inf{dim Rp |p ∈ V (I)}✳ ❱× ❝ị♥❣ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❞♦ ➤ã R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲ dim Rp = depth Rp ❤❛② ht I = inf {depth Rp |p ∈ V (I)} = depth(I, R) = dim R − dim R/I ❤❛② ht I + dim R/I = dim R✳ ✭✐✐✮ ❚❛ ➤➲ ❜✐Õt ✈➭♥❤ ➤❛ t❤ø❝ tr➟♥ ♠ét ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ✈➭♥❤ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✱ ❞♦ ➤ã t❛ ❝❤Ø ❝➬♥ ❝❤Ø r❛ s❛♦ ❝❤♦ R ❧➭ ❝❛t❡♥❛r②✳ ❚❤❐t ✈❐② ❧✃② p, q ∈ Spec R p ⊂ q✳ ❑❤✐ ➤ã Rq ❝ò♥❣ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤❡♦ ♠ô❝ ✭✐✮ t❛ ❝ã ht q = dim Rq = ht pRq + dim(Rq /pRq ) = ht p + ht q/p ❉♦ ➤ã R ❧➭ ❝❛t❡♥❛r②✳ download by : skknchat@gmail.com ✷✾ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺✳ ❬❘✷✱ ➜Þ♥❤ ❧Ý ✸✳✻❪ ●✐➯ sư r➺♥❣ R ❧➭ ♠✐Ị♥ ➤Þ❛ ♣❤➢➡♥❣✳ ❑❤✐ ➤ã ❤❛✐ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣❀ ✭✐✐✮ R ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✱ ✈➭♥❤ R/p ❝ị♥❣ ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✈➭ t❛ ❝ã ➤➻♥❣ t❤ø❝ dim R/p + dim Rp = dim R ✈í✐ ♠ä✐ p ∈ Spec(R)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ♠Ư♥❤ ➤Ị ♥➭② ❧➭ ❦Õt q✉➯ ❝đ❛ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✵ ✈➭ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✶✳ ✷✳✷✳ ❚Ý♥❤ ❝❛t❡♥❛r② ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ▼ơ❝ ♥➭② ❞➭♥❤ ➤Ĩ tr×♥❤ ❜➭② ❧➵✐ ❦Õt q✉➯ ❝đ❛ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪ ➤➢❛ r❛ ♥➝♠ ✶✾✾✾ ♥ã✐ r➺♥❣ ♥Õ✉ R ❧➭ ✈➭♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦➳♥ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ t❤× ♠ä✐ ➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥✱ ❦❤➠♥❣ ♥❤✃t t❤✐Õt ❧➭ ❣✐❛♦ ❤♦➳♥✱ ❧➭ ❝❛t❡♥❛r② ✈➭ ❝❤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❝ị♥❣ ➤ó♥❣ ♥Õ✉ R ❧➭ ♠✐Ị♥ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣✳ ❚r♦♥❣ ❦❤✉➠♥ ❦❤æ ❧✉❐♥ ✈➝♥ ♥➭② t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt Λ ❧➭ R✲➤➵✐ sè ❣✐❛♦ ❤♦➳♥✱ ❝ã ➤➡♥ ✈Þ✳ ❚r➢í❝ ❤Õt t❛ ❤Ư t❤è♥❣ ❧➵✐ ♠ét sè ❦Õt q✉➯ ✈Ị ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝✳ ❈❤♦ ✭R, m✮ ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ❜❛♦ ♥é✐ ①➵ ❝ñ❛ ❣✐➳ M M ❧➭ R✲♠➠➤✉♥✱ ❦Ý ❤✐Ö✉ ❊R (M ) ❧➭ ✈➭ ❍im (M ) ❧➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ i ✈í✐ ▼ét R✲ ✈í✐ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ K✱ m ❝đ❛ M ó ột í t ợ ị ĩ ♥❤➢ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✶✳ ♠➠➤✉♥ K ❈❤♦ ✭R, m✮ ị ó ề ợ ọ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ❝ñ❛ d✳ R ♥Õ✉ K ⊗R R ∼ = Hom(❍dm (R), ❊R (R/m)) ❈❤♦ (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝ã ❝❤✐Ị✉ d s❛✉ ➤➞② ❧➭ ♠ét sè ❦Õt q✉➯ ✈Ị ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ➤➢ỵ❝ trÝ❝❤ tr♦♥❣ ❬❆✱ ▼Ư♥❤ ➤Ị ✶✳✺✱ ✶✳✻✱ ➜Þ♥❤ ❧ý ✹✳✷✱ ❍Ư q✉➯ ✹✳✸❪✳ download by : skknchat@gmail.com ✸✵ ❇ỉ ➤Ị ✷✳✷✳✷✳ ❬❆❪ ✭✐✮ K ❧➭ ❞✉② ♥❤✃t s❛✐ ❦❤➳❝ ♠ét ➤➻♥❣ ❝✃✉ ✈➭ ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝ã ❝❤✐Ị✉ ❧➭ d✳ ✭✐✐✮ ◆Õ✉ R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ị rst S tì t ó K = ExtrS (R, S)✱ tr♦♥❣ ➤ã r = dim S − dim R ❧➭ sè ♥❣✉②➟♥ ♥❤á ♥❤✃t t❤á❛ ♠➲♥ ExtrS (R, S) = 0✳ ❱× t❤Õ✱ ✈í✐ ♠ä✐ p ∈ SuppR K t❛ ❝ã Kp ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ❝ñ❛ ✭✐✐✐✮ ❈❤♦ Rp ✳ R ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ T ❧➭ ♠ét R✲♠➠➤✉♥ ✈➭ S ❧➭ ♠ét R✲➤➵✐ sè ❤♦➭♥ t♦➭♥ ♣❤➻♥❣✱ ❝ã ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝✳ ❑❤✐ ➤ã✱ ♥Õ✉ t➽❝ ❝đ❛ T ⊗R S ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ S t❤× T ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ❝đ❛ R✳ ✭✐✈✮ ●✐➯ sư r➺♥❣ R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥ S ✳ ➜➷t KR = ExtnS (R, S); (n = dim S − dim R) ❑❤✐ ➤ã K(Rp ) ∼ = (KR )p ✈í✐ ♠ä✐ p ∈ SuppR KR ✳ ✭✈✮ ●✐➯ sö r➺♥❣ dim R ≥ ✈➭ ❝❤♦ a1 , a2 ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝♦♥ ❝đ❛ R✳ ❑❤✐ ➤ã ❞➲② a1 , a2 ❧➭ KR ✲❝❤Ý♥❤ q✉②✳ ❇ỉ ➤Ị ✷✳✷✳✸✳ ❬●◆❪ ●✐➯ sư r➺♥❣ (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥✱ ❈❤♦ Λ ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ P ∈ Spec(Λ) ✈➭ ➤➷t p = P ∩ R, Γ = Λ/P ✳ ❑❤✐ ➤ã ♥Õ✉ dim Γ ≥ 2✱ t❤× tå♥ t➵✐ ♠ét ❞➲② ❦❤í♣ ♥❣➽♥ α β −→ Γ −→ X −→ Y −→ ❝➳❝ Γ✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ ♠ét ♣❤➬♥ tư t ∈ m\p t❤á❛ ♠➲♥ ♥❤÷♥❣ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✭✐✮ ✭✐✐✮ depthR (X) ≥ 2❀ t ❧➭ X ✲❝❤Ý♥❤ q✉② ✈➭ tY = (0)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ K = KR/p ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ❝ñ❛ R/p✳ ❚❛ ➤➷t X = HomR/p HomR/p (Γ, K) , K download by : skknchat@gmail.com ✸✶ ✈➭ ❦Ý ❤✐Ö✉ α:Γ→X ❧➭ ➤å♥❣ ❝✃✉ ❝❤Ý♥❤ t➽❝ ❝đ❛ ❤➵♥ s✐♥❤✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✾ ❝ã ✷✳✷✳✷✱ ✭✈✮ t❛ ❝ã dim R/p = dim Γ ≥ ❧➭ ❤÷✉ ❞♦ ➤ã t❤❡♦ ❇ỉ ➤Ị depthR X ≥ 2✳ ❱× t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✷✱ ✭✐✈✮ t❛ ❝ã t❤Ó ①❡♠ ➤å♥❣ ❝✃✉ R/p✲♠➠➤✉♥ Γ Γ✲♠➠➤✉♥✳ ❑❤✐ ➤ã X Rp ⊗R α Kp ∼ = K(Rp /pRp ) = Rp /pRp ❧➭ ♠ét ➤➻♥❣ ❝✃✉✳ ❉♦ ➤ã ❧➭ ①♦➽♥ tù ❞♦✳ ❑❤✐ ➤ã ➤➷t α Y = Coker α ♥➟♥ t❛ ❝ã ❧➭ ♠ét ➤➡♥ ❝✃✉ ✈× t❛ ❝ã ♠ét ❞➲② ❦❤í♣ ♥❣➽♥ β α −→ Γ −→ X −→ Y −→ ▼➷t ❦❤➳❝ t❛ ❝ã ➤Ò ✷✳✷✳✷✱ ✭✈✮ t❛ ❝ã Yp = (0)✱ tå♥ t➵✐ t ∈ m\p s❛♦ ❝❤♦ tY = (0) ❞♦ ➤ã t❤❡♦ ❇ỉ t ❧➭ R/p✲❝❤Ý♥❤ q✉② ❦Ð♦ t❤❡♦ ♣❤➬♥ tư t ❧➭ X ✲❝❤Ý♥❤ q✉②✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ✈➭♥❤ R ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ♣❤➯✐ ❣✐❛♦ ❤♦➳♥✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ✈➭♥❤ ♥❣✉②➟♥ tè ♥Õ✉ ✐➤➟❛♥ ✵ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ❚r♦♥❣ ❦❤✉➠♥ ❦❤æ ❧✉❐♥ ✈➝♥ ❧✉➠♥ ❣✐➯ t❤✐Õt ✈➭♥❤ ❧➭ ❣✐❛♦ ❤♦➳♥✱ ❝ã ➤➡♥ ✈Þ ♥➟♥ ♠ét ✈➭♥❤ ❧➭ ✈➭♥❤ ♥❣✉②➟♥ tè ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♥ã ❧➭ ♠ét ♠✐Ị♥ ♥❣✉②➟♥✳ ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❧➭ ❦Õt q✉➯ ♠✃✉ ❝❤èt ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ❝❤Ý♥❤ ❝đ❛ ❬●◆❪✳ ▼Ư♥❤ ➤Ị ✷✳✷✳✹✳ ✈➭ ❣✐➯ sö r➺♥❣ ➤➵✐ ❬●◆❪ ❈❤♦ (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ Λ ❧➭ ♠ét ✈➭♥❤ ♥❣✉②➟♥ tè✳ ❑❤✐ ➤ã✱ ♥Õ✉ Λ ❝❤ø❛ ♠ét ✐➤➟❛♥ ❝ù❝ Q ✈➭ htΛ Q = t❤× dim Λ = ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ ✶✳✶✳✾ t❛ ❝ã R ❧➭ ♠ét ♠✐Ị♥ ♥❣✉②➟♥ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ t❛ ①Ðt tr➢ê♥❣ ❤ỵ♣ sư r➺♥❣ R dim Λ ≥ 2✳ ❣✐÷❛ ❝➳❝ f :R→Λ ❧➭ ➤➡♥ ❝✃✉✳ ❚❤❡♦ ▼Ư♥❤ ➤Ò dim R = dim Λ✳ ➜➬✉ t✐➟♥ ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥✳ ●✐➯ ❑❤✐ ➤ã t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✸ t❛ ❝ã ♠ét ♣❤Ð♣ ♥❤ó♥❣ Λ→X Λ✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ ♠ét ♣❤➬♥ tư = t ∈ m t❤á❛ ♠➲♥ ♥❤÷♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ depthR X ≥ ✈➭ ✭✷✮ t ❧➭ X ✲❝❤Ý♥❤ q✉② ✈í✐ tX ⊆ Λ download by : skknchat@gmail.com ✸✷ ❚❛ ➤➷t ❝ã IX ❞♦ Z = X/tX ∆ = ❊♥❞Λ Z ❈❤♦ I = ((0) :Λ Z) ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✷✮ ⊆ tX ⊆ Λ ✈➭ t❤❡♦ ❣✐➯ t❤✐Õt ❝ã Q ∈ Max Λ ♥➟♥ I ⊆ I X ⊆ tΛ ⊆ Q tΛ ⊆ I ⊆ Q ✶✳✷✳✻ ✈➭ t∈ / P ∩R Q/I ∈ Min Λ/I ✶✳✷✳✼ t❛ ❝ã ❱× t ❧➭ Λ✲❝❤Ý♥❤ ✈í✐ ❜✃t ❦× ❞♦ ➤ã q✉② ♥➟♥ P ∈ Min Λ✳ t ❦❤➠♥❣ ❧➭ ➢í❝ ❝đ❛ ▼➭ t❤❡♦ ❣✐➯ t❤✐Õt Q/I ∈ Min Λ/I ∩ Max Λ/I ✳ 0✱ t❤❡♦ ❇ỉ ➤Ị ♥➟♥ htΛ Q = ❱× t❤Õ ➳♣ ❞ơ♥❣ ❇ỉ ➤Ị depthR Λ/I = ▼➷t ❦❤➳❝ tõ ➤å♥❣ ❝✃✉ ❝❤Ý♥❤ t➽❝ λ:Λ→ ❊♥❞∆ Z ❝➯♠ s✐♥❤ ♣❤Ð♣ ♥❤ó♥❣ Λ/I → ❊♥❞Λ Z ♥➟♥ depthR Z = depthR X −1 > ❞♦ ➤ã depthR Λ/I > 0✱ ♠➞✉ t❤✉➱♥✳ ❇➞② ❣✐ê ❝❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✳ ❈❤♦ ✈➭♥❤ ➤➬② ➤ñ t❤❡♦ t➠ ♣➠ Q = R ⊗R Q R ❧➭ m✲❛❞✐❝ ❝đ❛ R✳ ❱× Q ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ Λ ♥➟♥ ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝đ❛ Λ = R ⊗R Λ✳ ❚❤❡♦ ❍Ư q✉➯ ✶✳✷✳✶✵ t❛ ❝ã htΛ Q = htΛ Q = 1✳ ❉♦ ➤ã ✈í✐ P ∈ Min Λ t❤× htΛ/P Q/P = 1✳ ❱× R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ♠ét ✈➭♥❤ ●♦r❡♥st❡✐♥✱ ➤Þ❛ ♣❤➢➡♥❣ ♥➟♥ ➜➷t dim Λ/P = p = P ∩ R t❛ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉✿ ✭✶✮ P ∩ Λ = ✈➭ ✭✷✮ p ∈ Min R✳ ❚❤❐t ✈❐②✱ t❛ ❝ã P ∩ Λ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ Λ ✈➭ Q ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ Λ ♥➟♥ Q ∩ Λ = Q✳ Q ⊇ P ∩ Λ ⊃ (0) t❤✉➱♥ ✈í✐ ♠➭ htΛ Q = htΛ/P Q/P = 1✳ ➤➡♥ ❝✃✉ ♥➟♥ ❉♦ ➤ã✱ ♥Õ✉ ❉♦ ➤ã s✉② r❛ ❚õ ➤ã ▼➷t ❦❤➳❝ Q⊇P ♠➞✉ f :R→Λ ❧➭ p ∩ R = (0)✳ ❑Ý ❤✐Ư✉ Q(R) ❧➭ tr➢ê♥❣ t❤➢➡♥❣ ❝đ❛ R✳ ❑❤✐ ➤ã✱ ✈× P Λp ∈ Max Λp ∩ Min Λp ➤Ò ✶✳✷✳✼ ❝ã Q = P ∩ Λ✳ P ∩ Λ = (0)✳ Rp ⊗R Λ ∼ = Rp ⊗Q(R) (Q(R) ⊗R Λ) ✈í✐ P ∩ Λ = (0) t❤× ①Ý❝❤ ♥❣✉②➟♥ tè ♥➟♥ t❛ ❝ã Λp ❧➭ ♠ét Rp ✲♠➠ ➤✉♥ tù ❞♦ ✭t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✺ ✈➭ ✶✳✶✳✾✮✳ ❉♦ ➤ã t❤❡♦ ❇æ dim Rp = 0✳ ❚❤❡♦ ❦❤➻♥❣ ➤Þ♥❤ ✭✷✮ ✈➭ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺✱ t❛ ❝ã dim R = dim R/p = dim Λ/P = ♠➭ dim Λ = dim R ♥➟♥ dim Λ = ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ❞ơ♥❣ ✈➭ ❬●◆❪ ●✐➯ sư r➺♥❣ R ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❛t❡♥❛r② ♣❤ỉ Λ ❧➭ ♠ét ✈➭♥❤ ♥❣✉②➟♥ tè✳ ❑❤✐ ➤ã t❛ ❝ã ➤➻♥❣ t❤ø❝ dim Λ = dim Λ/Q + htΛ Q download by : skknchat@gmail.com ✸✸ ✈í✐ ♠ä✐ Q ∈ Spec Λ ❈❤ø♥❣ ♠✐♥❤✳ ❈ã t❤Ĩ ❣✐➯ sư Q ∈ Max Λ✳ ➜➷t f : R → Λ ❧➭ ➤➡♥ ❝✃✉✳ ❚r➢í❝ ❤Õt ét trờ ợ h = ht Q ị ú ➤Õ♥ h − 1✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✹ t❛ ❝ã t❤Ĩ ❣✐➯ sư ❈❤♦ (0) = P0 ⊂ P1 ⊂ P2 ⊂ ⊂ Ph = Q ❧➭ ♠ét ①Ý❝❤ ❜➲♦ ❤♦➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✹ ➤➵✐ ❝đ❛ Λp ✈➭ h ≥ ✈➭ dim Λ/Ph−1 = 1✳ Λ✳ ➜➷t ▼➷t ❦❤➳❝ ✈× htΛp (Ph−1 Λp ) = htΛ Ph−1 = h − p = Ph−1 ∩ R✳ ❑❤✐ Ph−1 Λp ❧➭ ✐➤➟❛♥ ❝ù❝ ✭t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✵ ✮ ♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣✱ ✈➭♥❤ Λp ❝ã ❝❤✐Ò✉ ❑r✉❧❧ ❜➺♥❣ h − 1✳ ❚õ ▼Ư♥❤ ➤Ị ✶✳✶✳✾ ✈➭ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺ t❛ ❝ã dim Λ = dim R = dim R/p + dim Rp = dim Λ/Ph−1 + dim Λp = + (h − 1) = h ❇➞② ❣✐ê ❝❤♦ ♥➟♥ Q ∈ Spec Λ ✈➭ ➤➷t dim Λq = htΛq QΛq = htΛ Q q = Q ∩ R ❱× QΛq ∈ Max Λq ❉♦ ➤ã dim Rq = htΛ Q ▼➷t ❦❤➳❝ ✈× dim Λ/Q = dim R/q ✈➭ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺ t❛ ❝ã dim Λ = dim R = dim R/q + dim Rq ♥➟♥ dim Λ = dim Λ/Q + htΛ Q ❙❛✉ ➤➞② ❧➭ ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❜➭✐ ❜➳♦ ❝❤♦ t❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ♠ét R✲➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠➠➤✉♥ ✈➭ ✈➭♥❤ R ➜Þ♥❤ ❧ý ✷✳✷✳✻✳ ❧➭ ❬●◆✱ ➜Þ♥❤ ❧ý ✶✳✶❪ ●✐➯ sư R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✱ Λ R✲➤➵✐ sè ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ R✲♠➠➤✉♥ ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ❣✐❛♦ ❤♦➳♥✮✳ ❑❤✐ ➤ã Λ ❧➭ ❝❛t❡♥❛r② ♥Õ✉ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ P ⊂Q ❧➭ ❝➷♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ ♠ä✐ ①Ý❝❤ ♥❣✉②➟♥ tè ❜➲♦ ❤♦➭ ❣✐÷❛ q✉❛ ①Ðt ✈➭♥❤ Λ/P ♣❤➢➡♥❣ ❤♦➳ t➵✐ ❈❤♦ ✈➭ P t❛ ❝ã t❤Ĩ ❣✐➯ sư ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ Q ❝ã ❝ï♥❣ ➤é ❞➭✐✳ ❇➺♥❣ ❝➳❝❤ ❝❤✉②Ó♥ P = (0)✳ ➜➷t q = Q ∩ R✱ ❜➺♥❣ ❝➳❝❤ ➤Þ❛ q t❛ ❝ã t❤Ĩ ❣✐➯ sư t❤➟♠ R ❧➭ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ Q ∈ Max Λ✳ (0) = P0 ⊂ P1 ⊂ ⊂ Pn = Q ♥❣✉②➟♥ tè ❝ñ❛ Λ✳ Λ✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ ❧➭ ♠ét ①Ý❝❤ ❜➲♦ ❤♦➭ ❝➳❝ ✐➤➟❛♥ n = dim Λ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ download by : skknchat@gmail.com n✳ ❚❤❡♦ ✸✹ ❣✐➯ t❤✐Õt R ➤ó♥❣ ✈í✐ n = 1✳ ❑❤✐ ➤ã tõ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✹ t❛ ❝ã ❦❤➻♥❣ ➤Þ♥❤ ❚❛ ❝ã t❤Ĩ ❣✐➯ sư dim Λ/P1 = n − ✈➭ n≥2 ✈➭ ❦❤➻♥❣ ➤Þ♥❤ ❧➭ ➤ó♥❣ ➤Õ♥ htΛ P1 = 1✱ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✺ t❛ ❝ã n = dim Λ/P1 + = dim Λ/P1 + htΛ P1 = dim Λ✳ ✈í✐ n − 1✳ ❱❐② ❦❤➻♥❣ ➤Þ♥❤ ➤ó♥❣ n✱ ❤❛② Λ ❧➭ ❝❛t❡♥❛r②✳ ❍Ư q✉➯ ✷✳✷✳✼✳ ●✐➯ sư ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ R ❧➭ ♠ét ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ Λ ❧➭ ♠ét R✲➤➵✐ sè R✲♠➠ ➤✉♥✳ ❑❤✐ ➤ã Λ ❧➭ ❝❛t❡♥❛r②✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜➞② ❧➭ ❤Ö q✉➯ trù❝ tế ủ ị ý í ì ột ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❍Ư q✉➯ ✷✳✷✳✽✳ ♠ét ❬●◆❪ ●✐➯ sư (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r✱ Λ ❧➭ R✲♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❑❤✐ ➤ã Λ ❧➭ ❝❛t❡♥❛r② ✈➭ t❛ ❝ã ❝➠♥❣ t❤ø❝ dim Λ = dim Λ/Q + htΛ Q ✈í✐ ♠ä✐ ✐➤➟❛♥ Q ∈ Spec(Λ) ✈➭ ➤➻♥❣ t❤ø❝ n = htΛ Q − htΛ P ➤ó♥❣ ✈í✐ ♠ä✐ ❝➷♣ P ⊆ Q ❝ñ❛ Λ ✈➭ ♠ä✐ ①Ý❝❤ ❜➲♦ ❤♦➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè P = P0 ⊂ P1 ⊂ ⊂ Pn = Q ❣✐÷❛ P ✈➭ ◗✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã t❤Ĩ ❣✐➯ sư ❝ã ➤å♥❣ ❝✃✉ ♥❤ó♥❣ R →Λ f :R→Λ ❝đ❛ ❝➳❝ ❧➭ ➤➡♥ ❝✃✉✳ ➜➷t R✲♠➠➤✉♥✳ ❱× Λ d = dim R✳ ❧➭ ❘✲♠➠ ➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ♥➟♥ Λ ❝ò♥❣ ❧➭ R✲♠➠ ➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ dimR Λ AssR R ⊆ AssR Λ ❤❛② ✈➭ dim R/p = d ✈í✐ ♠ä✐ ❚❛ = d✳ ❉♦ ➤ã p ∈ AssR R ❬▼❛✷✱ tr❛♥❣ ✶✵✼❪✱ R/p ị tự trộ ì t❤Õ R/p ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ❑❤✐ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺ t❛ ❝ã R/p ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✱ ❦Ð♦ t❤❡♦ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✻ t❛ ❝ã Λ ❧➭ ❝❛t❡♥❛r②✳ ▲✃② Q ∈ Min Λ ▼Ư♥❤ ➤Ị ✶✳✷✳✻✱ ✈➭ ➤➷t q = Q ∩ R✳ ❚❛ sÏ ❝❤Ø r❛ dim Λ/Q = d ❚❤❡♦ q ❜❛♦ ❣å♠ ❝➳❝ ➢í❝ ❝đ❛ tr♦♥❣ Λ✳ ❉♦ ➤ã q ⊆ p∈AssR Λ p ❬▼❛✷✱ tr❛♥❣ ✺✵❪✳ ❚❤❡♦ ➜Þ♥❤ ❧ý tr➳♥❤ ♥❣✉②➟♥ tè✱ tå♥ t➵✐ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè p ∈ AssR Λ s❛♦ ❝❤♦ q ⊆ p✳ ❱× dim R/p = d ✈➭ Λ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ download by : skknchat@gmail.com ✸✺ dimR Λ = d ♥➟♥ q = p ✈➭ p ∈ Min R✳ ❉♦ ➤ã dim R/q = d ♠➭ Λ/Q ❧➭ R/q ♠➠➤✉♥ ❜ë✐ ✈❐② t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✾ t❛ ❝ã dim Λ/Q = d ✈í✐ ♠ä✐ Q ∈ Min Λ✳ ◆❤ê tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ R✱ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✻ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✺ t❛ ❝ã t❤Ĩ ❞Ơ ❞➭♥❣ t❤✃② ➤➢ỵ❝ r➺♥❣ ❜✃t ❦× ♠ét ❝➷♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ Λ ✈í✐ ♥➺♠ ❣✐÷❛ t❤ø❝ P P ∈ Min Λ ✈➭ ✈➭ Q ∈ Max Λ✱ ①Ý❝❤ ❜➲♦ ❤♦➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ❝ï♥❣ ➤é ❞➭✐ ✈➭ ❜➺♥❣ Q dim Λ = dim Λ/Q + htΛ Q n = htΛ Q − htΛ P d = dim Λ✳ ➤ó♥❣ ✈í✐ ♠ä✐ ➤ó♥❣ ✈í✐ ♠ä✐ ❝➷♣ P ⊆ Q ❉♦ ➤ã t❛ ❝ã ❝➠♥❣ Q ∈ Spec Λ✳ ❚❛ ❝ò♥❣ ❝ã P ⊆ Q ✈➭ ♠ä✐ ①Ý❝❤ ❜➲♦ ❤♦➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè P = P0 ⊂ P1 ⊂ ⊂ Pn = Q ❣✐÷❛ P ✈➭ ◗ tr♦♥❣ Λ✳ ❉♦ ➤ã ❜ỉ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❑Õt q✉➯ s❛✉ ❧➭ ❤Ư q✉➯ trù❝ t✐Õ♣ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✻ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✺ ✈➭ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺ ❍Ư q✉➯ ✷✳✷✳✾✳ ❈❤♦ R ❧➭ ♠ét ♠✐Ị♥ ♥❣✉②➟♥ ➤Þ❛ ♣❤➢➡♥❣ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✳ ❑❤✐ ➤ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣✳ ✭✐✐✮ R ❧➭ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✳ ✭✐✐✐✮ ▼ä✐ ♠ä✐ R✲➤➵✐ sè ♥❣✉②➟♥ tè✱ ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠➠ ➤✉♥ Λ ❧➭ ❝❛t❡♥❛r② ✈➭ ✈í✐ Q ∈ Spec Λ t❛ ❝ã ➤➻♥❣ t❤ø❝ dim Λ = dim Λ/Q + htΛ Q download by : skknchat@gmail.com ✸✻ ❑Õt ❧✉❐♥ ❚ã♠ ❧➵✐✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ s❛✉✳ ✶✳ ◆❤➽❝ ❧➵✐ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ➤å♥❣ t❤ê✐ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ tÝ♥❤ ❝❤✃t ❧✐➟♥ q✉❛♥ ➤Õ♥ ♠ë ré♥❣ ♥❣✉②➟♥ tr➟♥ ♠ét ✈➭♥❤✱ ❝➳❝ ➤Þ♥❤ ý s s ợ ị ý tr ị ❧ý ❣♦✐♥❣✲✉♣✱ ✳✳✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ✈Ý ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ✷✳ ◆❤➽❝ ❧➵✐ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ị tÝ♥❤ ❝❛t❡♥❛r②✱ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ✈➭♥❤✳ ✸✳ ❚r×♥❤ ❜➭② ❧➵✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ❦Õt q✉➯ ❝❤Ý♥❤ tr♦♥❣ ❜➭✐ ❜➳♦ ✧❈❆❚❊◆❆❘■❚❨ ■◆ ▼❖❉❯▲❊✲❋■◆■❚❊ ❆▲●❊❇❘❆❙✧ ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②❝ñ❛ ❙✳ ●♦t♦ ✈➭ ❑✳ ◆✐s❤✐❞❛ ❬●◆❪ ✈Ị tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✳ download by : skknchat@gmail.com ✸✼ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❚✐Õ♥❣ ❆♥❤ ❬❆❪ ❨✳ ❆♦②❛♠❛ ✭✶✾✽✸✮✱ ❙♦♠❡ ❜❛s✐❝ r❡s✉❧ts ♦♥ ❝❛♥♦♥✐❝❛❧ ♠♦❞✉❧❡s✱ ❏✳ ▼❛t❤✳ ❑②♦t♦ ❯♥✐✈✳ ✷✸ ✱ ♥♦✳ ✶✱ ✽✺✲✾✹✳ ▼❘ ✻✾✷✼✸✶ ✭✽✹✐✿✶✸✵✶✺✮ ❬❆▼❪ ▼✳ ❋✳ ❆t✐②❛❤ ❛♥❞ ■✳ ●✳ ▼❛❝❉♦♥❛❧❞ ✭✶✾✻✾✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✱ ❆❞❞✐s♦♥✲❲❡s❧❡② P✉❜❧✐s❤✐♥❣ ❈♦✳✱ ❘❡❛❞✐♥❣✱ ▼❛ss✳✲▲♦♥❞♦♥✲❉♦♥ ▼✐❧❧s✱ ❖♥t✳✱ ✶✾✻✾✳ ▼❘ ✵✷✹✷✽✵✷ ✭✸✾ ✲ ✹✶✷✾✮ ❬❇❍❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣ ✭✶✾✾✸✮✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✸✾ ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✶✾✾✸✳ ▼❘ ✶✷✺✶✾✺✻ ✭✾✺❤✿✶✸✵✷✵✮ ❬❈✶❪ ■✳ ❙✳ ❈♦❤❡♥ ✭✶✾✹✻✮✱ ❖♥ t❤❡ str✉❝t✉r❡ ❛♥❞ ✐❞❡❛❧ t❤❡♦r② ♦❢ ❝♦♠♣❧❡t❡ ❧♦❝❛❧ r✐♥❣s✱❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✺✾ ✱ ✺✹✲✶✵✻✳ ▼❘ ✵✵✶✻✵✾✹ ✭✼✱✺✵✾❤✮ ❬❈✷❪ ■✳ ❙✳ ❈♦❤❡♥ ✭✶✾✺✹✮✱ ▲❡♥❣t❤ ♦❢ ♣r✐♠❡ ✐❞❡❛❧ ❝❤❛✐♥s✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✼✻ ✱ ✻✺✹✲✻✻✽✳ ▼❘ ✵✵✻✷✶✶✻ ✭✶✺✱✾✷✾❢✮ ❬●◆❪ ❙✳ ●♦t♦ ❛♥❞ ❑✳ ◆✐s❤✐❞❛ ✭✶✾✾✾✮✱ ❈❛t❡♥❛r✐t② ✐♥ ♠♦❞✉❧❡✲❢✐♥✐t❡ ❛❧❣❡❜r❛s Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❛♠❡r✐❝❛♥ ♠❛t❤❡♠❛t✐❝❛❧ s♦❝✐❡t②✳✶✷✼✱ ♥♦✳ ✶✷✱ ♣♣✳ ✸✹✾✺✲ ✸✺✵✷ ❬❍❑❪ ❏✳ ❍❡r③♦❣ ❛♥❞ ❊✳ ❑✉♥③ ✭✶✾✼✶✮✱ ❉❡r ❑❛♥♦♥✐s❝❤❡ ▼♦❞✉❧ ❡✐♥❡s ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✲❘✐♥❣s✱ ▲❡❝➢t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✷✸✽✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ◆❡✇ ❨♦r❦✱ ❚♦❦②♦✱ ✶✾✼✶✳ ▼❘ ✺✹✿✸✵✹ ❬❑❪ ■✳ ❑❛♣❧❛♥s❦② ✭✶✾✼✵✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣s✱ ❆❧❧②♥ ❛♥❞ ❇❛❝♦♥✱ ❇♦st♦♥✳ ▼❘ ✹✵✿✼✷✸✹ download by : skknchat@gmail.com ✸✽ ❬❑r❪ ❲✳ ❑r✉❧❧ ✭✶✾✸✼✮✱ ❩✉♠ ❉✐♠❡♥s✐♦♥s❜❡❣r✐❢❢ ❞❡r ■❞❡❛❧t❤❡♦r✐❡✱ ▼❛t❤✳ ❩✳✱ ✹✷ ✱ ♣♣✳ ✼✹✺✲✼✻✻✳ ❬▼❛✶❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✻✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✽✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❚r❛♥s❧❛t❡❞ ❢r♦♠ t❤❡ ❏❛♣❛♥❡s❡ ❜② ▼✳ ❘❡✐❞✳ ▼❘ ✽✼✾✷✼✸ ✭✽✽❤✿✶✸✵✵✶✮❢ ❬▼❛✷❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✵✮✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ✭s❡❝♦♥❞ ❡❞✐t✐♦♥✮✱ ▼❛t❤✲ ❡♠❛t✐❝s ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s✱ ✈♦❧✳ ✺✻ ✱ ❇❡♥❥❛♠✐♥✴❈✉♠♠✐♥❣s P✉❜❧✐s❤✐♥❣ ❈♦✳✱ ■♥❝✳✱ ❘❡❛❞✐♥❣✱ ▼❛ss✳ ▼❘ ✺✼✺✸✹✹ ✭✽✷✐✿✶✸✵✵✸✮ ❬▼❝❘❪ ❙✳ ▼❝❆❞❛♠ ❛♥❞ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✼✮✱ ❙❡♠✐✲❧♦❝❛❧ t❛✉t r✐♥❣s✱ ■♥❞✐❛♥❛ ❯♥✐✈✳ ▼❛t❤✳ ❏✳✷✻✱ ✼✸✲✼✾✳ ❬▼❘❪ ❏✳ ❈✳ ▼❝❈♦♥♥❡❧❧ ❛♥❞ ❏✳ ❈✳ ❘♦❜s♦♥ ✭✶✾✽✼✮✱ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ◆♦❡t❤❡r✐❛♥ r✐♥❣s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ✭◆❡✇ ❨♦r❦✮✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ▲t❞✳✱ ❈❤✐❝❤❡st❡r✱ ✶✾✽✼✳ ❲✐t❤ t❤❡ ❝♦♦♣❡r❛t✐♦♥ ♦❢ ▲✳ ❲✳ ❙♠❛❧❧❀ ❆ ❲✐❧❡②✲ ■♥t❡rs❝✐❡♥❝❡ P✉❜❧✐❝❛t✐♦♥✳ ▼❘ ✾✸✹✺✼✷ ✭✽✾❥✿✶✻✵✷✸✮ ❬◆✶❪ ▼✳ ◆❛❣❛t❛ ✭✶✾✺✻✮✱ ❖♥ t❤❡ ❝❤❛✐♥ ♣r♦❜❧❡♠ ♦❢ ♣r✐♠❡ ✐❞❡❛❧s✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✺✻ ✭✶✾✺✻✮✱ ✺✶✲✻✹✳ ▼❘ ✶✽✿✽❡ ❬◆✷❪ ▼✳ ◆❛❣❛t❛ ✭✶✾✻✷✮✱ ▲♦❝❛❧ r✐♥❣s✱ ■♥t❡rs❝✐❡♥❝❡ ❚r❛❝ts ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ◆♦✳ ✶✸ ✱ ▼❘ ✵✶✺✺✽✺✻ ✭✷✼ ✲ ✺✼✾✵✮ ❬❘✶❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✻✺✮✱ ❖♥ q✉❛s✐✲✉♥♠✐①❡❞ s❡♠✐✲❧♦❝❛❧ r✐♥❣s ❛♥❞ t❤❡ ❛❧t✐t✉❞❡ ❢♦r♠✉❧❛✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✽✼ ✱ ✷✼✽✲✷✽✹✳ ▼❘ ✸✶✿✸✹✹✽ ❬❘✷❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✻✾✮✱ ❖♥ q✉❛s✐✲✉♥♠✐①❡❞ ❧♦❝❛❧ ❞♦♠❛✐♥s✱ t❤❡ ❛❧t✐t✉❞❡ ❢♦r♠✉❧❛✱ ❛♥❞ t❤❡ ❝❤❛✐♥ ❝♦♥❞✐t✐♦♥ ❢♦r ♣r✐♠❡ ✐❞❡❛❧s ✭■✮✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✾✶ ✱ ✺✵✽✲✺✷✽✳ ▼❘ ✹✵✿✶✸✻ ❬❘✸❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✶✮ ✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❝❛t❡♥❛r② r✐♥❣s✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✾✸✱ ♣♣✳ ✶✵✼✵✲✶✶✵✽ ❬❙❤❪ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✾✵✮✱ ❙t❡♣ ✐♥ ❈♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✱ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ❙t✉❞❡♥t ❚❡①ts✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ download by : skknchat@gmail.com ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM VĂN CHUYỀN TÍNH CATENARY TRONG CÁC ĐẠI SỐ HỮU HẠN SINH NHƯ MÔĐUN Chuyên ngành: Đại số Lý thuyết số Mã số: 62.46.01.04 LUẬN VĂN THẠC SĨ... R S −1 p htR p = htS −1 R S −1 p ❚r➢ê♥❣ ợ ệt ế ọ S = Rp tì t htR p = htRp pRp = dim Rp ✶✳✷✳ số ữ s ột ỗ f : R → Λ ❧➭ ♠ét ➤å♥❣ ❝✃✉ ❣✐÷❛ ❝➳❝ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✳ a∈R ✈➭ b ∈ Λ✱ ✈í✐ ✈➠... R✲➤➵✐ sè✳ ◆❤➢ Λ ❝ï♥❣ ✈í✐ ♠ét ➤å♥❣ ❝✃✉ ✈➭♥❤ f : R → Λ✳ ❬❆▼❪ ▼ét ➤å♥❣ ❝✃✉ ✈➭♥❤ f : R ợ ọ ữ ợ ọ số ữ s ột R✲♠➠➤✉♥ ✭s❛✉ ➤➞② sÏ ❣ä✐ t➽t ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✮ ♥Õ✉ Λ ❤÷✉ ❤➵♥ s✐♥❤ ♥❤➢ ♠ét R✲♠➠➤✉♥✳