Đại học thái nguyên Tr-ờng đại học s- phạm nGuyễn minh thuận vành địa ph-ơng quy Luận văn thạc sĩ toán học Thái Nguyên - Năm 2013 S hóa Trung tâm Học liệu http://www.lrc-tnu.edu.vn/ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ợ trì tr t tr✉♥❣ t❤ù❝✱ ❝❤➢❛ ➤➢ỵ❝ sư ❞ơ♥❣ ❝❤♦ ❜➯♦ ✈Ư ♠ét ❤ä❝ ✈Þ ♥➭♦✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ sù ➤å♥❣ ý ❝đ❛ ❝➳❝ ❝➳ ♥❤➞♥ ✈➭ tæ ❝❤ø❝✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t➭✐ ệ trì tr ợ râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✾ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ◆❣✉②Ơ♥ ▼✐♥❤ ❚❤✉❐♥ ❳➳❝ ♥❤❐♥ ❝đ❛ tr➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❳➳❝ ♥❤❐♥ ❝đ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ➜♦➭♥ ❚r✉♥❣ ❈➢ê♥❣ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❝❤Ø ❜➯♦ ✈➭ ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ ❚❙✳ ➜♦➭♥ ❚r✉♥❣ ❈➢ê♥❣✳ ❚❤➬② ➤➲ ❞➭♥❤ ♥❤✐Ị✉ t❤ê✐ ❣✐❛♥ ❤➢í♥❣ ❞➱♥ ✈➭ ❣✐➯✐ ➤➳♣ ❝➳❝ t❤➽❝ ♠➽❝ ❝đ❛ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ ❜➬② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬②✳ ❚➠✐ ①✐♥ ❣ư✐ tí✐ ❝➳❝ t❤➬② ❝➠ ❑❤♦❛ ❚♦➳♥✱ ❑❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❝ị♥❣ ♥❤➢ ❝➳❝ t❤➬②✱ ❝➠ ë ✈✐Ư♥ t♦➳♥ ❤ä❝ ✲ ✈✐Ö♥ ❦❤♦❛ ❤ä❝ ✈➭ ❝➠♥❣ ♥❣❤Ö ❱✐Öt ◆❛♠ ➤➲ t❤❛♠ ❣✐❛ ❣✐➯♥❣ ❞➵② ❦❤ã❛ ❤ä❝ ✷✵✶✶✲✷✵✶✸✱ ❧ê✐ s s t ề ỗ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❣✐➳♦ ❞ơ❝✱ ➤➭♦ t➵♦ ❝đ❛ ♥❤➭ tr➢ê♥❣✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❙ë ◆é✐ ✈ô✱ ❙ë ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ▲➭♦ ❈❛✐✱ ❚r➢ê♥❣ ❚❍P❚ sè ✶ ▼➢ê♥❣ ❑❤➢➡♥❣✱ tæ ❚♦➳♥✲❚✐♥ ❚r➢ê♥❣ ❚❍P❚ sè ✶ ▼➢➡♥❣ ❑❤➢➡♥❣ ♥➡✐ t➠✐ ➤❛♥❣ ❝➠♥❣ t➳❝ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➠✐ ❤♦➭♥ t❤➭♥❤ ❦❤ã❛ ❤ä❝ ♥➭②✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ❜➵♥ ❜❒ ✈➭ ♥❣➢ê✐ t❤➞♥ ➤➲ q✉❛♥ t➞♠✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥✱ ➤é♥❣ ✈✐➟♥✱ ❝ỉ ✈ị ➤Ĩ t➠✐ ❝ã t❤Ĩ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ ❝đ❛ ♠×♥❤✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✾ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ◆❣✉②Ô♥ ▼✐♥❤ ❚❤✉❐♥ ✐✐✐ ▼ơ❝ ❧ơ❝ ▼ë ➤➬✉ ✶ ✶ ❱➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ✸ ✶✳✶ ❍Ư s✐♥❤ ❝ù❝ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ➜Þ♥❤ ♥❣❤Ü❛ ✈➭ ❝➳❝ ✈Ý ❞ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❍Ö t❤❛♠ sè ❝❤Ý♥❤ q✉② ✷ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ➜➷❝ tr➢♥❣ ➤å♥❣ ➤✐Ị✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ✶✺ ✷✳✶ ❈❤✐Ị✉ ①➵ ➯♥❤ ❝đ❛ ♠➠➤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❚Ý♥❤ ❝❤✃t ➤å♥❣ ➤✐Ị✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❱➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ❚Ý♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ✸✶ ✸✳✶ ❱➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❚Ý♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❑Õt ❧✉❐♥ ✹✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✸ ✶ ▼ë ➤➬✉ ❈❤♦ (R, m, k) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② ♥❤✃t m✳ ●ä✐ µ(m) ❧➭ sè ♣❤➬♥ tư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m✳ ❚❛ ❧✉➠♥ ❝ã µ(m) ≥ dim R ◆Õ✉ ➤➻♥❣ t❤ø❝ ①➯② r tứ à(m) = dim R, tì R ợ ❣ä✐ ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❑❤➳✐ ♥✐Ư♠ ✈➭♥❤ ị í q t ợ r ❜ë✐ ❲♦❧❢❛♥❣ ❑r✉❧❧ ✈➭♦ ♥➝♠ 1937✳ ❚✉② ♥❤✐➟♥ ♥ã ❝❤Ø t❤ù❝ sù ➤➢ỵ❝ q✉❛♥ t➞♠ tr♦♥❣ ❝✉è♥ s➳❝❤ ❝đ❛ ❖s❝❛r ❩❛r✐s❦✐ ✈➭✐ ♥➝♠ s❛✉ ➤ã✳ ❚r♦♥❣ ❝✉è♥ s➳❝❤ ➤ã ❩❛r✐s❦✐ ➤➲ ❝❤Ø r❛ ✈❛✐ trß ➤➷❝ ❜✐Ưt q✉❛♥ trä♥❣ ❝đ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② tr♦♥❣ ❤×♥❤ ❤ä❝ ➤➵✐ sè✳ ↕♥❣ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♠ét ➤✐Ó♠ tr➟♥ ♠ét ➤❛ t➵♣ ➤➵✐ sè ❧➭ ❦❤➠♥❣ ❦ú ❞Þ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈➭♥❤ ❝➳❝ ❤➭♠ ❝❤Ý♥❤ q✉② t➵✐ ➤✐Ĩ♠ ➤ã ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❚õ ➤ã ❝➳❝ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ➤➲ ➤➢ỵ❝ r✃t ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ♥❣❤✐➟♥ ❝ø✉ ❝ị♥❣ ♥❤➢ t×♠ ❤✐Ĩ✉ ❝➳❝ ø♥❣ ❞ơ♥❣ tr♦♥❣ ➤➵✐ sè✱ ❧ý t❤✉②Õt sè ✈➭ ❤×♥❤ ❤ä❝ ➤➵✐ sè✳ ❙ù r❛ ➤ê✐ ❝đ❛ ➤➵✐ sè ➤å♥❣ ➤✐Ị✉ ➤➲ ❜ỉ s✉♥❣ t❤➟♠ ♠ét ❝➠♥❣ ❝ơ ♠í✐ ➤➷❝ ❜✐Ưt ❤÷✉ Ý❝❤ ❝❤♦ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉②✳ ❈➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❆✉s❧❛♥❞❡r✲ ❇✉❝❤s❜❛✉♠✲ ❙❡rr❡ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❧➭ ❝❤Ý♥❤ q✉② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ä✐ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ➤Ị✉ ❝ã ❝❤✐Ị✉ ①➵ ➯♥❤ ❤÷✉ ❤➵♥✳ ❉ù❛ ✈➭♦ ➤ã ❤ä ➤➲ ❝❤ø♥❣ ♠✐♥❤ sù ❜➯♦ t♦➭♥ q✉❛ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ tÝ♥❤ ❝❤Ý♥❤ q✉② ♠ét ❝➳❝❤ ➤➡♥ ❣✐➯♥ ✭▼ét ➤Þ♥❤ ❧ý ♠➭ tr➢í❝ ➤✃② ❝❤ø♥❣ ♠✐♥❤ ❤Õt sø❝ ♣❤ø❝ t➵♣✮✳ ➜å♥❣ t❤ê✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❆✉s❧❛♥❞❡r✲ ❇✉❝❤s❜❛✉♠✲ ◆❛❣❛t❛ ❝ò♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ♠ét tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ❦❤➳❝ ❝đ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❧➭ tÝ♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ trì tết ị ĩ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q ợ trì tr ủ ❏❛②❛♥✲ t❤❛♥✱ ❘❡❣✉❧❛r ❧♦❝❛❧ r✐♥❣ ✭✷✵✵✺✮✳ ❈✃✉ tró❝ ❝đ❛ ❧✉❐♥ ó ợ trì ụ tể s ợ ù ể trì ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè ❦Õt q✉➯ ✈Ị ✈➭♥❤ í q tr trờ ợ ị r ó tết ợ ể ị ột số ế tứ ✈Ị ❤Ư s✐♥❤ ❝ù❝ t✐Ĩ✉ ✈➭ sè ♣❤➬♥ tư s✐♥❤ ự tể ủ ết ợ ù ể trì ❜➭② ➤Þ♥❤ ♥❣❤Ü❛✱ ♠ét sè tÝ♥❤ ❝❤✃t ✈➭ ✈Ý ❞ơ í q ị ột tr ữ ết q í ủ ợ trì tr tết ➜Þ♥❤ ❧ý ✶✳✷✳✾ ♥ã✐ r➺♥❣ ♠ä✐ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Þ❛ ♣❤➢➡♥❣ ➤Ị✉ ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✳ ❚✐Õt ✸ ➤➢ỵ❝ ❞➭♥❤ ➤Ĩ tr×♥❤ ❜➭② ✈Ị ❤Ư t❤❛♠ sè ❝❤Ý♥❤ q✉② ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ t✐Õt ✸ ❧➭ ➤➷❝ tr➢♥❣ tÝ♥❤ ❝❤Ý♥❤ q✉② ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② q✉❛ tÝ♥❤ ❝❤✃t ❝ñ❛ ✈➭♥❤ ♣❤➞♥ ❜❐❝ ❧✐➟♥ ❦Õt✳ ❚✐Õ♣ t❤❡♦ tr♦♥❣ ❝❤➢➡♥❣ ✷✱ ❝❤ó♥❣ t➠✐ sÏ tr×♥❤ ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t ➤å♥❣ ➤✐Ị✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ➜Ĩ ❝❤✉➮♥ ❜Þ ❝❤♦ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ❝❤ó♥❣ t➠✐ sÏ ❞➭♥❤ t✐Õt ✶ ➤Ĩ ♥❤➽❝ ❧➵✐ ➤Þ♥❤ ♥❣❤Ü❛✱ ❧✃② ✈Ý ❞ơ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ✈Ị ❝❤✐Ị✉ ①➵ ➯♥❤ ❝đ❛ ♠➠➤✉♥✳ ❚r♦♥❣ t✐Õt ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭②✱ tr♦♥❣ ➤ã ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ ➤➷❝ tr➢♥❣ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❧➭ ❝❤Ý♥❤ q✉② t❤➠♥❣ q✉❛ tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ ❝❤✐Ị✉ ➤å♥❣ ➤✐Ị✉ ❝đ❛ ❝➳❝ ♠➠➤✉♥ tr➟♥ ➤ã ✭➜Þ♥❤ ❧ý ❆✉s❧❛♥❞❡r ✲ ❇✉❝❤s❜❛✉♠ ✲ ❙❡rr❡✮✳ ▼ét ❤Ư q✉➯ q✉❛♥ trä♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ♥➭② ❧➭ sù ❜➯♦ t♦➭♥ q✉❛ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ tÝ♥❤ í q ũ ợ trì tr tết ố ù ợ ể trì ề í q tr trờ ợ ị r tết ✶ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ♥➟✉ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ t✐Õt ✶ ❧➭ ♠Ư♥❤ ➤Ị ✸✳✶✳✸ ✈Ị sù t➢➡♥❣ ➤➢➡♥❣ ❣✐÷❛ tÝ♥❤ ❝❤Ý♥❤ q✉② ❝ñ❛ ♠ét ✈➭♥❤ ✈➭ ✈➭♥❤ ➤❛ t❤ø❝ tr➟♥ ➤ã r ó tết ợ ù ể trì ❜➭② ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❝❤➢➡♥❣✳ ❉ù❛ ✈➭♦ ❝➳❝ ❦Õt q✉➯ ➤➲ ❝ã ë ❝❤➢➡♥❣ ✶ ✈➭ ❝❤➢➡♥❣ ✷ ❝❤ó♥❣ t➠✐ sÏ ❝❤ø♥❣ ♠✐♥❤ ♠ä✐ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Ò✉ ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ✭➜Þ♥❤ ❧ý ❆✉s❧❛♥❞❡r ✲ ❇✉❝❤s❜❛✉♠ ✲ ◆❛❣❛t❛✮✳ ✸ ❈❤➢➡♥❣ ✶ ❱➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ❚r♦♥❣ s✉èt ❧✉❐♥ ✈➝♥ ♥➭②✱ ♠ét ✈➭♥❤ ❧✉➠♥ ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✱ ❝ã ➤➡♥ ✈Þ ❦❤➳❝ ❦❤➠♥❣✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❈❤➢➡♥❣ ✶ ❧➭ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè ❦Õt q✉➯ ✈Ị ✈➭♥❤ ❝❤Ý♥❤ q✉② tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ị ết ợ ể ị ột sè ❦✐Õ♥ t❤ø❝ ✈Ị ❤Ư s✐♥❤ ❝ù❝ t✐Ĩ✉ ✈➭ sè ♣❤➬♥ tư s✐♥❤ ❝ù❝ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥✳ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ tr♦♥❣ t✐Õt ♥➭② sÏ ➤➢ỵ❝ ❞ï♥❣ ➤Ĩ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② tr♦♥❣ ❝➳❝ t✐Õt s❛✉✳ ❚r♦♥❣ t✐Õt ú t trì ị ĩ ột số tí ❝❤✃t ✈➭ ✈Ý ❞ơ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Þ❛ ♣❤➢➡♥❣✳ ▼ét tr♦♥❣ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ❧➭ ♠ä✐ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ➤Ị✉ ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✳ ❚✐Õt ợ ể trì ề ệ t số ❝❤Ý♥❤ q✉② ❝đ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Þ❛ ♣❤➢➡♥❣✳ ✶✳✶ ❍Ư s✐♥❤ ❝ù❝ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥ ❚❛ ❧✉➠♥ ①Ðt (R, m, k) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ✈➭ tr➢ê♥❣ t❤➷♥❣ ❞➢ k = R/m✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳ ▼ét ❤Ư s✐♥❤ ❜✃t ❦ú ♣❤➬♥ tư m x1 , x2 , , xn ❝ñ❛ m ❧➭ tè✐ t✐Ĩ✉ ♥Õ✉ t❛ ❜á ➤✐ xi ♥➭♦ t❤× ♣❤➬♥ ❝ß♥ ❧➵✐ x1 , x2 , , xi−1 , xi+1 , , xn ❦❤➠♥❣ ❧➭ ❤Ư s✐♥❤ ❝đ❛ m✳ ✹ ❱Ý ❞ơ ❈❤♦ ✶✳✶✳✷✳ k ❧➭ ♠ét tr➢ê♥❣✱ ①Ðt ✈➭♥❤ k[[X1 , , Xn ]]✳ ❚❛ ❜✐Õt k[[X1 , , Xn ]] ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ❝ã ✐➤➟❛♥ ❝ù❝ ➤➵✐ m = (X1 , , Xn )✳ ❉Ơ t❤✃② r➺♥❣ ❤Ư X1 , , Xn ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m✳ ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❝❤♦ t❛ ♠ét ➤✐Ị✉ ❦✐Ư♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ➤Þ♥❤ ♥❣❤Ü❛ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✸✳ ❈❤♦ x1 , x2 , , xn ∈ m✳ ❑❤✐ ➤ã ❤Ö x1 , x2 , , xn ❧➭ ❤Ö s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x1 , x2 , , xn ∈ R/m2 ❧➭ ❝➡ së ❝ñ❛ k ✲ ❦❤➠♥❣ ❣✐❛♥ ✈Ð❝t➡ m/m2 ✳ ❈❤ø♥❣ ♠✐♥❤✳ (⇒) ●✐➯ sö x1 , x2 , , xn ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m✳ ❙✉② r❛ x1 , x2 , , xn ❧➭ ❤Ư s✐♥❤ ❝đ❛ m/m2 ✳ ❚❤❐t ✈❐② ①Ðt ♠ét ♣❤➬♥ tö x ∈ m/m2 ✳ ❉♦ x ∈ m = (x1 , , xn ) ♥➟♥ n xi , ∈ R x= i=1 ❙✉② r❛ n n x= xi + m = i=1 tr♦♥❣ ➤ã n 2 (ai + m)(xi + m ) + m = i=1 xi , i=1 = + m ∈ R/m, xi = xi + m2 ∈ m/m2 ✳ ❱❐② x1 , x2 , , xn ❤Ư s✐♥❤ ❝đ❛ m/m2 ✳ ❇➞② ❣✐ê t❛ ❣✐➯ sö ❧➭ n x1 , x2 , , xn ❦❤➠♥❣ ➤é❝ ❧❐♣ t✉②Õ♥ ✈➭ x i , ∈ x1 = i=2 R/m s✉② r❛ n xi ∈ m2 = (x1 , x2 , , xn )2 x1 − i=2 ✺ n n ⇔ x1 − aij xi xj , aij ∈ R xi = i=2 n i,j=1 n ⇔ x1 − i=2 (i,j)=(1,1) n n ⇔ x1 (1 − a1 x1 ) = ⇔ x1 = x1 ∈ m ♥➟♥ ( x i + − a1 x1 i=2 − a1 x ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt aij xi xj xi + i=2 n ✭❱× aij xi xj xi = a11 (x1 ) + (i,j)=(1,1) n aij xi xj ) ∈ (x2 , , xn ) (i,j)=(1,1) ❦❤➯ ♥❣❤Þ❝❤ tr♦♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ x1 , x2 , , xn ❧➭ ❤Ö s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ R✮✳ ➜✐Ị✉ ♥➭② m✳ ❉♦ ➤ã ❤Ö x1 , x2 , , xn ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤✳ ❱❐② x1 , x2 , , xn ❧➭ ❝➡ së ❝ñ❛ k ✲ ❦❤➠♥❣ ❣✐❛♥ ✈Ð❝t➡ m/m2 ✳ (⇐) ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sö x1 , x2 , , xn ❧➭ ❝➡ së ❝ñ❛ k−❦❤➠♥❣ ❣✐❛♥ ✈Ð❝t➡ m/m2 ✳ ❚❛ ❝ã m = (x1 , x2 , , xn )R + m2 ♥➟♥ m/I = (m2 + I)/I = m(m/I), ✈í✐ I = (x1 , x2 , , xn )✳ ❚❤❡♦ ❜ỉ ➤Ị ◆❛❦❛②❛♠❛ t❛ ❝ã m/I = ❤❛② m = I ✳ ❱❐② x1 , x2 , , xn ❧➭ ❤Ư s✐♥❤ ❝đ❛ m✳ ❚✐Õ♣ t❤❡♦ t❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ❤Ö x1 , x2 , , xn ❧➭ tè✐ t✐Ĩ✉✳ ❚❤❐t ✈❐② ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐ ❤Ư ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ ❣✐❛♥ ✈Ð❝t➡ m✱ s✉② r❛ tå♥ t➵✐ ❤Ö xi1 , , xit ❧➭ ♠ét ❤Ö m✳ ❚❤❡♦ ❣✐➯ t❤✐Õt s✉② r❛ xi1 , , xit ❧➭ ❝➡ së ❝ñ❛ k ✲ ❦❤➠♥❣ m/m2 ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt x1 , x2 , , xn ❧➭ ❝➡ së ❝ñ❛ k ✲❦❤➠♥❣ ❣✐❛♥ ✈Ð❝t➡ m/m2 ✳ ❱❐② x1 , x2 , , xn ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✹✳ ❈❤♦ ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ ❞ơ ✶✳✶✳✺✳ ❈❤♦ m✳ (R, m, k) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r✳ ❙è ♣❤➬♥ tư ❝đ❛ ♠ét ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭ ❱Ý x1 , x2 , , xn µ(m)✳ k ❧➭ ♠ét tr➢ê♥❣✱ ①Ðt ✈➭♥❤ k[[X1 , , Xn ]]✳ ❚❛ ❜✐Õt k[[X1 , , Xn ]] ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ❝ã ✐➤➟❛♥ ❝ù❝ ➤➵✐ m = ✻ (X1 , , Xn ) ✈➭ ❤Ö X1 , , Xn ❧➭ ♠ét ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m✳ ❱❐② t❛ ❝ã µ(m) = n✳ ❇ỉ ➤Ị ✶✳✶✳✻✳ ❚❛ ❝ã ✭❛✮ µ(m) = dimk (m/m2 )✳ ✭❜✮ µ(m) ≥ dim(R)✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭❛✮ ●✐➯ sö x1 , x2 , , xn ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m ✈í✐ n ▼➷t ❦❤➳❝ t❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✸ ❤Ư = µ(m)✳ x1 , x2 , , xn ❧➭ ❝➡ së ❝ñ❛ k ✲❦❤➠♥❣ ❣✐❛♥ ✈Ð❝t➡ m/m2 ♥➟♥ t❛ ❝ã dimk (m/m2 ) = n✳ ❱❐② t❛ ❝ã µ(m) = dimk (m/m2 )✳ ✭❜✮ ●✐➯ sö x1 , x2 , , xn ❧➭ ❤Ö s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m✳ ❱× dim(R/(x1 , x2 , , xn )) = dim(R/m) = 0, ♥➟♥ tå♥ t➵✐ ♥➵♣ t❤❡♦ ✰ ❱í✐ xi s❛♦ ❝❤♦ dim(R/xi R) < dim R✳ dim R dim R = 0, ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ ✰ ●✐➯ sư dim R > 1✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ❣✐❛♦ ❑r✉❧❧ t❛ ❝ã dim(R/x1 R) = ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m ✭✈í✐ m ❧➭ ✐➤➟❛♥ dim R − 1✳ ▼➷t ❦❤➳❝ ✈× x2 , , xn ❝ñ❛ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② R/x1 R✮✳ ❇➺♥❣ q✉② ♥➵♣ ❝❤♦ R/x1 R t ợ à(m) dim(R/x1 R) à(m) ≥ dim R − ❱❐② t❛ ❝ã µ(m) ≥ dim(R)✳ ❚Ý♥❤ ❝❤✃t s❛✉ ➤➞② ❝đ❛ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ➤➢ỵ❝ ❞ï♥❣ ♥❤✐Ị✉ ❧➬♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✼✳ ➯♥❤ ✈í✐ ❈❤♦ I ⊆ m ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R✳ ◆Õ✉ x1 , x2 , , xr ∈ m ❝ã x1 , x2 , , xr ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ m/I t❤× tå♥ t➵✐ xr+1 , , xn ∈ I r ≤ n s❛♦ ❝❤♦ ❤Ö x1 , , xr , xr+1 , , xn ❧➭ ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝ñ❛ m✳ ✷✾ ∞ aij X i Y j ∈ S ✳ ❇➞② ❣✐ê t❛ t❤❛② Y = X ✈í✐ ✈➭♦ α ❝❤➻♥❣ ❤➵♥ ♥❤➢ (i,j)=0 a32 X Y = a32 X XY = a32 Y X a42 X Y = a42 X Y = a42 Y ❑❤✐ ➤ã t❛ ➤➢ỵ❝ s✐♥❤ ❜ë✐ α = a + bX + (Y − X )k[[X, Y ]] ✈í✐ a, b ∈ k[[Y ]]✳ ❱❐② S 1, X ✳ ▼➷t ❦❤➳❝ t❛ t❤✃② ❤Ư ✈í✐ 1, X ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤ t❤❐t ✈❐② ❣✐➯ sö a.1 + b.X = a, b ∈ k[[Y ]]✳ ❙✉② r❛ t❛ ❝ã a(Y ).1 + b(Y ).X ∈ (Y − X ) ⇔ a(Y ).1 + b(Y ).X = C(X, Y )(Y − X ) ❙✉② r❛ a(Y ).1 + b(Y ).X = C(X, Y ).Y − C(X, Y ).X ∞ ∞ i Ci (Y ).X i X Ci (Y )X Y − = i=0 i=0 = C0 (Y ).Y + C1 (Y ).Y.X + C2 (Y ).Y.X + C3 (Y ).Y.X + C4 (Y ).Y.X + · · · − C0 (Y ).X − C1 (Y ).X − C2 (Y ).X − · · · = C0 (Y ).Y + C1 (Y ).Y.X + (C2 (Y ).Y − C0 (Y ))X + (C3 (Y ).Y − C1 (Y ))X + (C4 (Y ).Y − C2 (Y )).X + ··· ➜å♥❣ ♥❤✃t ❤Ư sè ❤❛✐ ✈Õ ❝đ❛ ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ➤➢ỵ❝ t❛ ➤➢ỵ❝ ❤Ư s❛✉ a(Y ) = C0 (Y ).Y b(Y ) = C1 (Y ).Y C2 (Y ).Y − C0 (Y ) = C3 (Y ).Y − C1 (Y ) = C4 (Y ).Y − C2 (Y ) = ✸✵ ❚õ ➤ã s✉② r❛ a(Y ) = C0 (Y ).Y = C2 (Y ).Y = C4 (Y ).Y = C6 (Y )Y = = C2n (Y ).Y n+1 ●✐➯ sö ∞ bi Y i ⇒ a(Y ) = Y r (br + br+1 Y + · · · ), = a(Y ) = i=0 ✈í✐ br = 0✳ ❧ý✳ ❱❐② ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❤× a(Y ) ❝❤✐❛ ❤Õt ❝❤♦ b(Y ) = 0✳ ❱❐② 1, X r❛ ❧➭ ❤❛② ❝ã ❝➡ së ❧➭ S ➤✐Ò✉ ♥➭② ✈➠ a(Y ) = 0✳ ▼ét ❝➳❝❤ t➢➡♥❣ tù t❛ ❝ò♥❣ ❝ã S Y r+1 ❧➭ 1, X ❤❛② S R✲♠➠➤✉♥ ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤✳ ❙✉② tù ❞♦✳ ❱❐② S ❧➭ R✲♠➠➤✉♥ ①➵ ➯♥❤ R✲♠➠➤✉♥ ♣❤➻♥❣✳ ❈✉è✐ ❝ï♥❣ t❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ q✉②✳ ❚❤❐t ✈❐②✱ t❛ ❝ã S/mS = S/yS ∼ = k[[X]]/(X ) ❦❤➠♥❣ ❧➭ ❝❤Ý♥❤ dim k[[X]] = dim k + = 1✳ ♥❣✉②➟♥ tè ❝ã ❝❤✐Ò✉ ❝❛♦ ♥❤✃t tr♦♥❣ t❤❛♠ sè tr♦♥❣ ▼➷t ❦❤➳❝ k[[X]] ♠➭ (X) = 0✱ s✉② r❛ X ❧➭ ✐➤➟❛♥ ❧➭ ♣❤➬♥ tư k[[X]]✳ ❑❤✐ ➤ã t❛ ➤➢ỵ❝ dim(S/mS) = dim(k[[X]]/(X )) = − = ■➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ 1✳ ❱❐② t❛ ❝ã ❝❤Ý♥❤ q✉②✳ k[[X]]/(X ) ❧➭ (X)/(X ) = (X)✱ s✉② r❛ µ((X)/(X )) = dim(S/mS) = = = µ((X)/(X ))✱ ❤❛② S/mS ❦❤➠♥❣ ❧➭ ✸✶ ❈❤➢➡♥❣ ✸ ❱➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ❚Ý♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t❛ sÏ ➤✐ tì ể ề í q ị t❤ê✐ ♥❣❤✐➟♥ ❝ø✉ ♠ét tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❧➭ tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ▼ä✐ ✈➭♥❤ ➤➢ỵ❝ ①Ðt tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ➤Ị✉ ❧➭ ✈➭♥❤ ◆♦❡t❤❡r ✈➭ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ✸✳✶ ❱➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ ❝❤➢➡♥❣ tr➢í❝✱ sư ❞ơ♥❣ ❝➳❝ ➤➷❝ tr➢♥❣ ➤å♥❣ ➤✐Ị✉✱ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ❧➭ ♠ét ✈➭♥❤ ❝❤Ý♥❤ q✉② ✭➜Þ♥❤ ❧ý ✷✳✷✳✸✮✳ ❚õ ➤ã ❝❤♦ ♣❤Ð♣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ✭♥❤✃t t❤✐Õt✮ ➤Þ❛ ♣❤➢➡♥❣ s ị ĩ R ợ ọ ❝❤Ý♥❤ q✉② ♥Õ✉ ♥ã ❧➭ ✈➭♥❤ ◆♦❡t❤❡r ✈➭ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ t➵✐ ♠ä✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❱Ý ❞ô ✸✳✶✳✷✳ (1) ❳Ðt ✈➭♥❤ ❝➳❝ sè ♥❣✉②➟♥ Z✳ ❚❛ ➤➲ ❜✐Õt Z ❧➭ ✈➭♥❤ ◆♦❡t❤❡r✳ ●✐➯ sư pZ ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ Z✱ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ Z t➵✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè p t❛ ➤➢ỵ❝ Zp ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❱❐② Z ❧➭ ♠ét ✈➭♥❤ ❝❤Ý♥❤ q✉②✳ ✸✷ (2) ❚❛ ❝ã C[X1 , , Xn ] ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉②✳ ❚❤❐t ✈❐②✱ t❤ø ♥❤✃t t❛ ➤➲ ❜✐Õt r➺♥❣ C[X1 , , Xn ] ❧➭ ✈➭♥❤ ◆♦❡t❤❡r✳ ❚❤ø ❤❛✐✱ ♠ä✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ C[X1 , , Xn ] ➤Ò✉ ❝ã ❞➵♥❣ m = (X1 − ❑❤✐ ➤ã C[X1 , , Xn ]m ∼ = x1 , , Xn − xn )✱ ✈í✐ x1 , , xn ∈ C✳ C[Y1 , , Yn ](Y1 , ,Yn ) , Xi → Yi + xi ✱ ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉②✳ ❱❐② C[X1 , , Xn ] ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉②✳ ▼Ö♥❤ ➤Ò ✸✳✶✳✸✳ ❱➭♥❤ ◆♦❡t❤❡r R ❧➭ ❝❤Ý♥❤ q✉② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ R[X] ❧➭ ❝❤Ý♥❤ q✉②✳ ❈❤ø♥❣ ♠✐♥❤✳ (⇒) ●✐➯ sö R ❧➭ ❝❤Ý♥❤ q✉②✳ ●ä✐ M ❧➭ ♠ét ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ R[X] ✈➭ ➤➷t m = M ∩ R✳ ❚❛ ❝ã f (X) : f (X) ∈ Rm [X], g(X) ∈ / M} g(X) n f (X) i ={ : f (X) = X , ∈ R, si ∈ / m, g(X) ∈ / M} g(X) s i i=0 (Rm [X])M = { n bi X i f (X) ={ : f (X) = i=0 , bi ∈ R, s ∈ / m ⊂ M, g(X) ∈ / M} g(X) s h(X) ={ : h(X) ∈ R[X], sg(X) ∈ / M} sg(X) = R[X]M ❚õ ➤ã s✉② r❛ R[X]M ❧➭ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ tỉ♥❣ q✉➳t t❛ ❝ã t❤Ó t❤❛② ➜➷t Rm [X]✱ ❞♦ ✈❐② ❦❤➠♥❣ ♠✃t tÝ♥❤ R ❜➺♥❣ Rm ✳ k = R/m✳ ❚❛ ❝ã k[X] = (R/m)[X] ∼ = R[X]/m[X]✳ ❚❤❐t ✈❐②✱ ①Ðt ➤å♥❣ ❝✃✉ ϕ : R[X] → (R/m)[X], ✸✸ n X i → ①➳❝ ➤Þ♥❤ ❜ë✐ n X i ✳ ❚❛ t❤✃② ϕ ❧➭ t♦➭♥ ➳♥❤✱ ♠➷t ❦❤➳❝ t❛ ❝ã i=0 i=0 n n i ker ϕ = {f (X) = X i = 0} X | i=0 n i=0 X i |ai = 0, ∀i = 0, n} = {f (X) = i=0 n X i |ai ∈ m, ∀i = 0, n} = {f (X) = i=0 = m[X] ◆➟♥ s✉② r❛ ❚❛ ❝ã R[X]/m[X] ∼ = (R/m)[X]✳ R[X]/m[X] ❝ã ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❧➭ (R/m)[X] s❛♦ ❝❤♦ tr➢ê♥❣ ♥➟♥ R/m[X] tö s✐♥❤✮✳ ❙✉② r❛ m0 =< f (X) > M/m[X]✱ ✈í✐ s✉② r❛ tå♥ t➵✐ f (X) ∈ R[X] ✭❉♦ m0 ∈ R/m ❧➭ ❧➭ ♠✐Ò♥ ✐➤➟❛♥ ❝❤Ý♥❤ ♥➟♥ ♠ä✐ ✐➤➟❛♥ ➤Ò✉ ❝ã ♠ét ♣❤➬♥ M/m[X] =< f (X) > ❤❛② M = (m, f )✳ ❚❤❐t ✈❐② ❧✃② α ∈ M ⇒ α ∈ M/m[X]✱ s✉② r❛ α = bf (X) ⇔ α + m[X] = b(f (X) + m[X]), ❤❛② α = bf (X) + g(X) ✈í✐ g(X) ∈ m[X]✳ ❱❐② M = (m, f )✳ ❚✐Õ♣ t❤❡♦ t❛ ❝❤ø♥❣ ♠✐♥❤ ht M = + ht m✳ s✉② r❛ tå♥ t➵✐ ①Ý❝❤ ♥❣✉②➟♥ tè ❝ã ➤é ❞➭✐ m = p0 p1 ❚❤❐t ✈❐② ❣✐➯ sö t❛ ❝ã ht m = a a pa ▼➷t ❦❤➳❝ t❛ ❝ã m = R ∩ M ⇒ M ⊃ mR[X], t❛ ➤➢ỵ❝ ♠ét ①Ý❝❤ ❝➳❝ ✐➤➟❛♥ s❛✉ M mR[X] = p0 R[X] ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ❝ã ♠✐Ò♥ ♥❣✉②➟♥ ♥➟♥ p1 R[X] pa R[X](∗) R[X]/pi R[X] ∼ = (R/pi )[X] ♠➭ (R/pi )[X] ❧➭ R[X]/pi R[X] ❧➭ ♠✐Ò♥ ♥❣✉②➟♥✳ ❉♦ ➤ã pi R[X] ❧➭ ❝➳❝ ✐➤➟❛♥ ✸✹ ♥❣✉②➟♥ tè✱ ❤❛② (∗) ❧➭ ♠ét ①Ý❝❤ ♥❣✉②➟♥ tè ❝ñ❛ M✳ ❚õ ➤ã s✉② r❛ ht M ❤❛② = a+1 ht M = + ht m✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ dim R[X]M = + dim R✳ ❚❤❐t ✈❐② t❛ ❝ã ht M = max{n : M = p0 p1 dim R[X]M = max{r : MR[X]M ❉♦ ❝➳❝ Qi ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R[X]M pn } Q1 Qr } ♥➟♥ ❝➳❝ Qi ❝ã ❞➵♥❣ Qi = pi R[X]M s✉② r❛ r = n✳ ❱❐② t❛ ❝ã ht M = dim R[X]M ✳ ❈✉è✐ ❝ï♥❣ ❞♦ t❛ ❝ã t❤Ó t❤❛② R ❜➺♥❣ Rm ♥➟♥ t❤❡♦ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ❝ã ht m = dim R✳ ❱❐② t❛ ❝ã dim R[X]M = ht M = + ht m = + dim R ▼➷t ❦❤➳❝ ❞♦ R M = (m, f ) ❧➭ ❝❤Ý♥❤ q✉② ♥➟♥ ♥➟♥ M m s✐♥❤ ❜ë✐ sè ♣❤➬♥ tö ❜➺♥❣ dim R✱ ♠➭ s✐♥❤ ❜ë✐ sè ♣❤➬♥ tö ❜➺♥❣ dim R + 1✳ dim R[X]M = dim R + = µ(M)✳ ❱❐② R[X]M ❚õ ➤ã s✉② r❛ ❧➭ ❝❤Ý♥❤ q✉② ❤❛② R[X] ❧➭ ❝❤Ý♥❤ q✉②✳ (⇐) ●✐➯ sö R[X] ❧➭ ❝❤Ý♥❤ q✉②✳ ●ä✐ m ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ R ✈➭ ➤➷t n = (m, X) ❚➢➡♥❣ tù ♥❤➢ tr➟♥ t❛ ❝ã R[X]n ❧➭ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ ❧➭♠ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ t❤❛② Rm [X] ❞♦ ✈❐② ❦❤➠♥❣ R ❜➺♥❣ Rm ✳ ❚✐Õ♣ tơ❝ ❝ị♥❣ t❤❡♦ ❝➳❝ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ❝ò♥❣ ❝ã ❦Õt q✉➯ s❛✉ dim R[X]n = ht n = + ht m = + dim R(1) ❉♦ x ∈ n2 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x ∈ nR[X]n /(nR[X]n )2 ♥➟♥ tå♥ t➵✐ ♠ét ❤Ư s✐♥❤ tè✐ t✐Ĩ✉ ❝ã x1 , , xn ❝ñ❛ m s❛♦ ❝❤♦ (x1 , , xn ) ❧➭ ❤Ö s✐♥❤ tè✐ t✐Ĩ✉ ❝đ❛ n tõ ➤ã t❛ µ(n) = µ(m) + 1✳ R[X]n ▼➷t ❦❤➳❝ t❤❡♦ ❣✐➯ t❤✐Õt R[X] ❧➭ ❝❤Ý♥❤ q✉② ♥➟♥ s✉② r❛ ❧➭ ❝❤Ý♥❤ q✉②✱ tõ ➤ã t❛ ❝ã dim R[X]n = µ(nR[X]n ) = µ(n) = µ(m) + 1(2) ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ R ❧➭ ❝❤Ý♥❤ q✉②✳ + dim R = + µ(m)✳ ❱❐② t❛ ❝ã dim R = µ(m) ❤❛② ✸✺ ✸✳✷ ❚Ý♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ❝ñ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❚r♦♥❣ t✐Õt ♥➭② ❝❤ó♥❣ t❛ sÏ ❝❤ó♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ♠ä✐ ✈➭♥❤ ❝❤Ý♥❤ q✉② ➤Ò✉ ❧➭ ♠ét ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ▼ét ♠✐Ị♥ ♥❣✉②➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ♥Õ✉ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✶✳ ♠ä✐ ♣❤➬♥ tư ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❦❤➳❝ ➤➡♥ ✈Þ ➤Ị✉ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ♥❣✉②➟♥ tè✳ ❱Ý ❞ơ ✸✳✷✳✷✳ (1) ❱➭♥❤ ❝➳❝ sè ♥❣✉②➟♥ Z ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ (2) ❈➳❝ tr➢ê♥❣ sè ❤÷✉ tû Q✱ sè t❤ù❝ R ✈➭ sè ♣❤ø❝ C ➤Ò✉ ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ (3) ❈➳❝ ✈➭♥❤ Z[X1 , , Xn ]✱ Q[X1 , , Xn ]✱ R[X1 , , Xn ]✱ C[X1 , , Xn ], tr♦♥❣ ➤ã ❝➳❝ X1 , , Xn ❧➭ ❝➳❝ ❜✐Õ♥✱ ➤Ò✉ ❧➭ ❝➳❝ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ q✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ C[X] ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❚❛ ➤➲ ❜✐Õt ➤❛ t❤ø❝ f (x) = ax + b ❧➭ ♥❣✉②➟♥ tè tr♦♥❣ C[X] ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ < ax + b > ❧➭ ♥❣✉②➟♥ tè✳ ●✐➯ sö g(x).h(x) ∈< ax + b > ❦❤✐ ➤ã tÝ❝❤ ❝ñ❛ g(x) ✈➭ h(x) ❝ã ❞➵♥❣ g(x).h(x) = c(x)(ax + b) = c(x)ax + c(x)b ❚õ ➤ã t❛ ❝ã g(− ab ).h(− ab ) = s✉② r❛ b g(− ) = g(x) = k(x)(ax + b) a ⇔ ⇔ b h(x) = m(x)(ax + b) h(− ) = a ❱❐② < ax + b > ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ g(x) ∈< ax + b > h(x) ∈< ax + b > ◆❣➢ỵ❝ ❧➵✐ t❛ ①Ðt Spec C[X] = {0, < ax + b >: a = 0} ✸✻ ▲✃② p ∈ Spec C[X], p = 0✳ ●✐➯ sö f (x) ∈ p ✈➭ ❧➭ ➤❛ t❤ø❝ ❦❤➳❝ ✵ ✈➭ ❝ã ❜❐❝ ♥❤á ♥❤✃t✳ ▲✃② g(x) ∈ p s✉② r❛ g(x) = q(x)f (x) + r(x), deg r(x) < deg f (x) ❑❤✐ ➤ã t❛ ➤➢ỵ❝ r(x) = g(x) − q(x)f (x) ∈ p✳ ❉♦ ❝➳❝❤ ❝❤ä♥ f (x) ♥➟♥ r(x) = 0✳ ❱❐② ✈í✐ ♠ä✐ g(x) ∈ p t❤× g(x) ❝❤✐❛ ❤Õt ❝❤♦ f (x) ♥➟♥ s✉② r❛ p =< f (x) >✳ C[X] t❛ ❝ã f (x) = a(x − α1 )n1 (x − αr )nr ∈ p✳ ❱× p ❧➭ ▼➷t ❦❤➳❝ tr♦♥❣ ♥❣✉②➟♥ tè ♥➟♥ s✉② r❛ tå♥ t➵✐ ♥❤✃t ♥➟♥ s✉② r❛ (4) ❈❤♦ k x − αi s❛♦ ❝❤♦ x − αi ∈ p ♠➭ f (x) ❝ã ❜❐❝ ♥❤á f (x) = a(x − αi ) ❤❛② t❛ ❝ã p = (x − αi )✳ ❧➭ ♠ét tr➢ê♥❣✱ ❦❤✐ ➤ã k[X1 , , Xn ], tr♦♥❣ ➤ã ❝➳❝ X1 , , Xn ❧➭ ❝➳❝ ❜✐Õ♥✱ ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ q✉② ♥❤✃t✳ (5) ❈❤♦ k ❧➭ ♠ét tr➢ê♥❣✱ ❦❤✐ ➤ã k[[X1 , , Xn ]], tr♦♥❣ ➤ã ❝➳❝ X1 , , X n ❧➭ ❝➳❝ ❜✐Õ♥ ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ q✉② ♥❤✃t✳ ❚✐Õ♣ t❤❡♦ ❝❤ó♥❣ t❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ❜❛ ❜ỉ ➤Ị s❛✉ ➤Ĩ ♣❤ơ❝ ✈ơ ❝❤♦ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭②✳ ❇ỉ ➤Ị ✸✳✷✳✸✳ ❈❤♦ R ❧➭ ✈➭♥❤✱ S ❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ ❝ñ❛ R✳ ❑❤✐ ➤ã t❛ ❝ã Spec(S −1 R) = {pS −1 R, p ∩ S = ∅, p ∈ Spec(R)} ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö p ∈ Spec R, p ∩ S = ∅✳ ❑❤✐ ➤ã t❛ ❝ã a pS −1 R = { , a ∈ p, b ∈ S} b ●✐➯ sö dc fe r❛ tå♥ t➵✐ ❉♦ r❛ ∈ pS −1 R k∈S ✈í✐ dc , fe s❛♦ ❝❤♦ ∈ S −1 R s✉② r❛ dc fe k(ceb − adf ) = ∈ p = ✭❱× a b ✈í✐ p a ∈ p, b ∈ S ❧➭ ✐➤➟❛♥ ♥➟♥ s✉② ∈ p✮✳ S ∩ p = ∅ ♥➟♥ t❛ ❝ã k ∈ / p✱ s✉② r❛ ceb − adf ∈ p✳ ❚❛ ❧➵✐ ❝ã adf ∈ p✱ s✉② ceb ∈ p✳ ❑Õt ❤ỵ♣ ✈í✐ b ∈ S ce ∈ p ⇒ t❛ ➤➢ỵ❝ c ∈ pS −1 R c∈p ⇔ de e∈p ∈ pS −1 R f ✸✼ ❱❐② pS −1 R ∈ Spec S −1 R✳ ◆❣➢ỵ❝ ❧➵✐ ❧✃② Q ∈ Spec S −1 R✳ ➜➷t p = Q ∩ R = {a ∈ R : ●✐➯ sö a ∈ Q} x.y ∈ p ✈í✐ x, y ∈ R✱ s✉② r❛ x ∈Q x y 1 ∈Q⇒y ⇔ 1 ∈Q ❱❐② x∈p y∈p p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ Q = pS −1 R✳ ❚❤❐t ✈❐② t❛ ❝ã a b a a a ∈ Q ⇒ ∈ Q ⇒ ∈ Q ⇒ a ∈ p ⇒ ∈ pS −1 R ⇒ Q ⊆ pS −1 R b b 1 b ▲✃② dc ∈ pS −1 R, c ∈ p, d ∈ S, ❞♦ c a c ∈ Q ⇒ ∈ Q ⇒ ∈ Q ⇒ pS −1 R ⊆ Q 1 d d c∈p⇒ ❱❐② t❛ ❝ã Q = pS −1 R✳ ▼Ư♥❤ ➤Ị ✸✳✷✳✹✳ ▼ét ♠✐Ị♥ ♥❣✉②➟♥ ◆♦❡t❤❡r R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ ❝đ❛ ❈❤ø♥❣ ♠✐♥❤✳ (⇒) ●✐➯ sư R ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ✈➭ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ ❝ñ❛ s✉② r❛ a= ❝ñ❛ ♥➟♥ tå♥ t➵✐ R R ➤Ò✉ ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ i∈I R✳ ▲✃② a ∈ p ⊂ R✳ ❉♦ R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ♥➟♥ xi ✱ i tr♦♥❣ ➤ã ❝➳❝ s❛♦ ❝❤♦ xi x i ∈ p✱ ❧➭ ♥❣✉②➟♥ tè✳ ❉♦ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè t❛ ❣✐➯ sö ❚❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ x ∈ p✳ (x1 ) = p✳ ❚❤❐t ✈❐② ❍✐Ĩ♥ ♥❤✐➟♥ t❛ ❝ã ◆❣➢ỵ❝ ❧➵✐ ❞♦ (x1 ) ⊆ p✳ (x1 ) ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ✈➭ (x1 ) ⊆ p ♥➟♥ s✉② r❛ tå♥ t➵✐ ♠ét ①Ý❝❤ ♥❣✉②➟♥ tè ❝ã ❞➵♥❣ (x1 ) ⊆ p✳ ❚✉② ♥❤✐➟♥ t❛ ❧➵✐ ❝ã ht p = ♥➟♥ (x1 ) = p✳ (⇐) ●✐➯ sö ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ ❝ñ❛ R ➤Ị✉ ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ ✸✽ ❚r➢í❝ ❤Õt t❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ♠✐Ị♥ ♥❣✉②➟♥ ◆♦❡t❤❡r R ♠ä✐ ♣❤➬♥ tư ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❦❤➳❝ ➤➡♥ ✈Þ ➤Ị✉ ❝ã t❤Ĩ ♣❤➞♥ tÝ❝❤ t❤➭♥❤ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉②✳ ❚❤❐t ✈❐②✱ ①Ðt t❐♣ ❝➳❝ ✐➤➟❛♥ S = {Ra : ❛ ❦❤➠♥❣ ❧➭ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉②} ●✐➯ sö S = ∅✳ ❉♦ R ❧➭ ✈➭♥❤ ◆♦❡t❤❡r ♥➟♥ tå♥ t➵✐ ♠ét ♣❤➬♥ tư ❝ù❝ ➤➵✐ Ra0 ✈í✐ a0 ❦❤➠♥❣ ❧➭ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉②✳ ❱× a0 ❦❤➠♥❣ ❧➭ ❜✃t ❦❤➯ q✉② ♥➟♥ t❛ ❣✐➯ sư a0 = a1 a2 ✈í✐ a1 , a2 ❦❤➠♥❣ ❦❤➯ ♥❣❤Þ❝❤✳ ❉♦ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉② ♥➟♥ t❛ ❝ã ❤♦➷❝ ❜✃t ❦❤➯ q✉② ❤♦➷❝ a2 ❝ù❝ ➤➵✐ tr♦♥❣ R S✳ ❦❤➠♥❣ ❧➭ tÝ❝❤ ❦❤➠♥❣ ❧➭ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❦❤➠♥❣ ❧➭ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉②✳ ❑❤➠♥❣ ❧➭♠ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❣✐➯ sö r➺♥❣ q✉②✱ ❦❤✐ ➤ã t❛ ❝ã a1 a0 Ra1 ∈ S ❱❐② ✈➭ S=∅ a1 Ra0 ❦❤➠♥❣ ❧➭ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ Ra1 ✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ Ra0 ❧➭ ❤❛② ♥ã✐ ❝➳❝❤ ❦❤➳❝ tr♦♥❣ ♠✐Ị♥ ♥❣✉②➟♥ ◆♦❡t❤❡r ♠ä✐ ♣❤➬♥ tư ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❦❤➳❝ ➤➡♥ ✈Þ ➤Ị✉ ❝ã t❤Ĩ ♣❤➞♥ tÝ❝❤ ➤➢ỵ❝ t❤➭♥❤ tÝ❝❤ ❝đ❛ ❝➳❝ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉②✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ♠ä✐ ♣❤➬♥ tö ❜✃t ❦❤➯ q✉② tr♦♥❣ tö ♥❣✉②➟♥ tè✳ ❚❤❐t ✈❐② ❣✐➯ sư ➢í❝ ♥❣✉②➟♥ tè ♥❤á ♥❤✃t ❝đ❛ ❉♦ ➤ã a∈R a✳ R ❧➭ ♣❤➬♥ tư ❜✃t ❦❤➯ q✉② ❝đ❛ ➤Ị✉ ❧➭ ♣❤➬♥ R✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ✈Ị ✐➤➟❛♥ ❝❤Ý♥❤ s✉② r❛ ●ä✐ p ❧➭ ht p = 1✳ p ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✱ ❞♦ ✈❐② tå♥ t➵✐ b ∈ R s❛♦ ❝❤♦ p = (b)✳ ❚õ ➤ã t❛ ❝ã a = rb ✈í✐ r ∈ R✳ ❉♦ a ❧➭ ❜✃t ❦❤➯ q✉② ♥➟♥ r ❧➭ ➤➡♥ ✈Þ✳ ❱❐② (a) = (b) ❤❛② a ❧➭ ♣❤➬♥ tư ♥❣✉②➟♥ tè✳ ❇ỉ ➤Ị ✸✳✷✳✺✳ tr♦♥❣ ❈❤♦ R ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ◆♦❡t❤❡r✱ x ❧➭ ♠ét ♣❤➬♥ tư ♥❣✉②➟♥ tè R✳ ❑❤✐ ➤ã Rx ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ R ❝ò♥❣ ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ (⇒) ●✐➯ sư Rx ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ➜➷t S = {1, x, x2 , }✳ ●ä✐ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ tr♦♥❣ R s❛♦ ❝❤♦ p ∩S = ∅✳ ❑❤✐ ➤ã pRx ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ tr♦♥❣ Rx ✭t❤❡♦ ❇ỉ ➤Ị ✸✳✷✳✸✮✳ ❚❤❡♦ ❣✐➯ t❤✐Õt Rx ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ♥➟♥ t❤❡♦ ♠Ư♥❤ ➤Ị ✸✳✷✳✹✱ tå♥ ✸✾ t➵✐ a ∈ p s❛♦ ❝❤♦ pRx = aRx ✳ ❚❛ ❝❤ä♥ a s❛♦ ❝❤♦ (a) ⊂ R ❧➭ ❧í♥ ♥❤✃t t❤♦➯ ♠➲♥ tÝ♥❤ ❝❤✃t ➤➲ ❝❤♦✳ ❚❛ t❤✃② a ∈ (x)✱ s✉② r❛ a = bx, b ∈ R✳ ❉♦ a ∈ / (x)✳ a∈p s✉② r❛ ❚❤❐t ✈❐②✱ ❣✐➯ sö ♥❣➢ỵ❝ ❧➵✐ bx ∈ p ♠➭ p ♥❣✉②➟♥ tè ✈➭ p ∩ S = ∅ ⇒ b ∈ p✳ ❚õ ➤ã t❛ ❝ã (a) ⊂ (b) ⊂ p s✉② r❛ aRx ⊂ bRx ⊂ pRx ❱× ♥➟♥ t❛ ❝ã aRx = pRx ❱❐② bRx = pRx ✱ ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❝➳❝❤ ❝❤ä♥ a✳ (a) = (x)✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ❍✐Ó♥ ♥❤✐➟♥ t❛ ❝ã p = (a)✳ (a) ⊆ p✳ ◆❣➢ỵ❝ ❧➵✐ ❧✃② y∈p⇒ s✉② r❛ tå♥ t➵✐ xt s❛♦ ❝❤♦ ab y ∈ pRx = aRx = { n , b ∈ R}, x xt (yxn − ab) = 0✳ yxn = ab ∈ (a) x=0 ♥➟♥ s✉② r❛ s✉② r❛ (yxn−1 − ab1 )x = 0✳ yxn−1 = ab1 ∈ (a)✳ ❱❐② t❛ ❝ã ♠➭ (a) ❉♦ R ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ✈➭ ❧➭ ♥❣✉②➟♥ tè ♥➟♥ t❛ ❝ã ❚✐Õ♣ tô❝ ❞♦ R b = b1 x ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ♥➟♥ t❛ ❝ã ❈ø t✐Õ♣ tơ❝ ♥❤➢ ✈❐② t❛ ➤➢ỵ❝ y = abn s✉② r❛ y ∈ (a)✳ p = (a) ❤❛② p ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✳ ❱❐② R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ✭t❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✷✳✹✮✳ (⇐) ●✐➯ sư R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ▼ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ tr♦♥❣ Rx ❝ã ❞➵♥❣ pRx ✈í✐ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ✶ tr♦♥❣ R ✭❇ỉ ➤Ị ✸✳✷✳✸✮✳ ❑❤✐ ➤ã t❛ ❝ã pRx = { ❱❐② p = (a) ✈í✐ a ∈ p✳ ❚❛ ❝ã a b a a ba , n ≥ 0, b ∈ R} = { n , n ≥ 0, b ∈ R} = {r , r ∈ R} = ( ) n x x 1 pRx ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤ ❤❛②Rx ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t ✭t❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✷✳✹✮✳ ❈✉è✐ ❝ï♥❣ ❝❤ó♥❣ t❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý s❛✉✱ t❤➢ê♥❣ ❣ä✐ ❧➭ ➤Þ♥❤ ❧ý ❆uslander − Buchbaum − N agata✳ ✹✵ ➜Þ♥❤ ❧ý ✸✳✷✳✻✳ ✭➜Þ♥❤ ❧ý ❆✉s❧❛♥❞❡r ✲ ❇✉❝❤❜❛✉♠ ✲◆❛❣❛t❛✮ ▼ä✐ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ➤Ò✉ ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ dim R✳ ◆Õ✉ dim R = 0✱ ❦❤✐ ➤ã R ❧➭ ♠ét tr➢ê♥❣ ♥➟♥ R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ◆Õ✉ dim R = 1✱ ❝❤Ý♥❤ ♥➟♥ ●✐➯ sö ♥➟♥ ❦❤✐ ➤ã ❧➭ ✈➭♥❤ ➤Þ♥❤ ❣✐➳ rê✐ r➵❝✳ ❉♦ ➤ã R ❧➭ ♠✐Ò♥ ✐➤➟❛♥ R ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ dim R > 1✳ R/x R ▲✃② = x ∈ m/m2 ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ❤❛② ❝❤ø♥❣ ♠✐♥❤ x ❦❤✐ ➤ã R/xR ❧➭ ♥❣✉②➟♥ tè tr♦♥❣ R✳ ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❚❤❡♦ ❇ỉ ➤Ị ✸✳✷✳✺ ➤Ĩ R ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t t❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ Rx ❧➭ ♠✐Ò♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ●✐➯ sö p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ❝❤✐Ò✉ ❝❛♦ ✶ tr♦♥❣ Rx ✳ ➜➷t p = p ∩ R✱ ❦❤✐ ➤ã t❛ ➤➢ỵ❝ p = pRx ✳ ❚õ R ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ✈➭ p ❧➭ R✲ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✷✱ s✉② r❛ p ❝ã ❣✐➯✐ tù ❞♦ ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ❧➭ → Fn → Fn−1 → → F0 → p → 0(1), ✈í✐ ❝➳❝ tè ❝ñ❛ Fi Rx ✱ ❧➭ ❝➳❝ R✲♠➠➤✉♥ tù ❞♦ ❤÷✉ ❤➵♥ s✐♥❤✳ ◆Õ✉ P t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❉♦ ➤ã ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ (Rx )P = R(R∩P) ❧➭ p (Rx )P ❧➭ ✐➤➟❛♥ ❝❤Ý♥❤✱ t❛ ❝ã proj dim p = sup(proj dim p (Rx )P ) = 0, ♥❣❤Ü❛ ❧➭ p ❧➭ ♠ét ♠➠➤✉♥ ①➵ ➯♥❤✳ ❚✐Õ♣ t❤❡♦ t❛ t❡♥①➡ ✭✶✮ ✈í✐ Rx t❛ ➤➢ỵ❝ ❞➲② ❦❤í♣ → Fn ⊗ Rx → Fn−1 ⊗ Rx → → F0 ⊗ Rx → p ⊗ Rx → ❚❛ ➤➢ỵ❝ ❞➲② ❦❤í♣ d dn−1 d ϕ n → Fn → Fn−1 → →1 F0 → p → 0(2), tr♦♥❣ ➤ã Fi = Fi ⊗ Rx ❧➭ ❝➳❝ R✲♠➠➤✉♥ tù ❞♦ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ p ⊗ Rx = p ✳ P❤➞♥ tÝ❝❤ ❞➲② ✭✷✮ t❤➭♥❤ ❝➳❝ ❞➲② ❦❤í♣ ♥❣➽♥ → K0 → F0 → p → 0, → ✹✶ K1 → F1 → K0 → 0, , → Fn → Fn−1 → Kn−1 → 0(3) tr♦♥❣ ➤ã K0 = Im ϕ, Ki = Im di , ∀i = 1, n✳ ❳Ðt ❞➲② ❦❤í♣ → K0 → F0 → p → ❉♦ p ❧➭ ①➵ ➯♥❤ ♥➟♥ ❞➲② ♥➭② ❧➭ ❝❤❰ r❛✱ s✉② r❛ F0 = p ⊗ K0 ✳ ❱× F0 ❧➭ tù ❞♦ ♥➟♥ Ki F0 ❧➭ ①➵ ➯♥❤✱ s✉② r❛ K0 ❧➭ ①➵ ➯♥❤✳ ❍♦➭♥ t♦➭♥ t➢➡♥❣ tù ❝❤ó♥❣ t❛ ❝ã ❝➳❝ ➤Ị✉ ❧➭ ❝➳❝ ♠➠➤✉♥ ①➵ ➯♥❤✳ ❉♦ ➤ã ❝➳❝ ❞➲② ❦❤í♣ ♥❣➽♥ tr♦♥❣ ✭✸✮ ❧➭ ❝❤❰ r❛✱ s✉② r❛ Fn ∼ = n−❧❰ ❑❤✐ ➤ã sÏ tå♥ t➵✐ ❝➳❝ rank G = r✱ r❛ ❞♦ rank p = 1✳ Fn ⊗ p n−❝❤➼♥ Rx ✲♠➠➤✉♥ tù ❞♦ F ✈➭ G s❛♦ ❝❤♦ F ∼ = G⊗p✳ ●✐➯ sö p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❦❤➳❝ ❦❤➠♥❣ ❝đ❛ ♠✐Ị♥ ♥❣✉②➟♥ Rx s✉② ❱❐② t❛ ➤➢ỵ❝ rank F = r + 1✳ ▲✃② r+1 tÝ❝❤ ♥❣♦➭✐ ❝ñ❛ F Λr+1 F ∼ = Λr+1 (G ⊗ p )✳ ▼➷t ❦❤➳❝ t❛ ❝ã Λr+1 F = Rx ✈➭ Λi p = 0, ∀i > s✉② r❛ t❛ ❝ã Rx ∼ = p ✳ ❱❐② p ❧➭ tù ❞♦ ❤❛② p ❧➭ ❝❤Ý♥❤✳ ❙✉② ✈➭ G+p r❛ Rx t❛ ➤➢ỵ❝ ❧➭ ♠✐Ị♥ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ❚❤❡♦ ❇ỉ ➤Ị tÝ❝❤ ❞✉② ♥❤✃t✳ 3.2.5 t❛ ➤➢ỵ❝ R ❧➭ ♠✐Ị♥ ♣❤➞♥ ✹✷ ❑Õt ❧✉❐♥ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ➤➲ tr×♥❤ ❜➭② ❧➵✐ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❜➭✐ ❣✐➯♥❣ ❝ñ❛ ❆✳ ❱✳ ❏❛②❛♥t❤❛♥✱ ❘❡❣✉❧❛r ❧♦❝❛❧ r✐♥❣✱ ❆❞✈❛♥❝❡❞ ■♥tr✉❝t✐♦♥❛❧ ❙❝❤♦♦❧ ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❛♥❞ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ✭✹✲✸✵ ❏✉❧②✱ ✷✵✵✺✮✳ ▲✉❐♥ ✈➝♥ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ✿ ✶✳ ◆❤➽❝ ❧➵✐ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ❝đ❛ ❤Ư s✐♥❤ ❝ù❝ t✐Ĩ✉ ❝đ❛ ✐➤➟❛♥✳ ◆➟✉ ➤Þ♥❤ ♥❣❤Ü❛✱ ❧✃② ✈Ý ❞ơ ✈➭ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t ♠✐Ị♥ ♥❣✉②➟♥ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ◆➟✉ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư t❤❛♠ sè ❝❤Ý♥❤ q✉② ➳♣ ❞ơ♥❣ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➷❝ ❝❤➢♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② t❤➠♥❣ q✉❛ tÝ♥❤ ❝❤✃t ❝ñ❛ ✈➭♥❤ ♣❤➞♥ ❜❐❝ ❧✐➟♥ ❦Õt✳ ✷✳ ◆❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ①➵ ➯♥❤ ❝đ❛ ♠➠➤✉♥✱ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ❝đ❛ ❝❤✐Ị✉ ①➵ ➯♥❤ ➤Ĩ ❝❤✉➮♥ ❜Þ ❝❤♦ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t ➤å♥❣ ➤✐Ị✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉②✳ ❈❤ø♥❣ ♠✐♥❤ ➤➷❝ tr➢♥❣ ➤å♥❣ ➤✐Ị✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② t❤➠♥❣ q✉❛ ❝❤✐Ị✉ ➤å♥❣ ➤✐Ị✉ ✈➭ tÝ♥❤ ❝❤✃t ❜➯♦ t♦➭♥ ❝đ❛ tÝ♥❤ ❝❤Ý♥❤ q✉② q✉❛ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳✳ ✸✳ ➜Þ♥❤ ♥❣❤Ü❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ♠ä✐ ✈➭♥❤ ❝❤Ý♥❤ q✉② ❧➭ tÝ♥❤ ♣❤➞♥ tÝ❝❤ ❞✉② ♥❤✃t✳ ✹✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬❇❍❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣✱ ❈♦❤❡♥ ✲ ▼❛❝❛✉❧❛② ❘✐♥❣✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✸✳ ❬❏❛❪ ❆✳ ❱✳ ❏❛②❛♥t❤❛♥✱ ✧❘❡❣✉❧❛r ❧♦❝❛❧ r✐♥❣ ✧✱ ❆❞✈❛♥❝❡❞ ■♥str✉❝t✐♦♥❛❧ ❙❝❤♦♦❧ ♦♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❛♥❞ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ■♥❞✐❛♥✱ ✹✲✸✵ ❏✉❧②✱ ✷✵✵✺✳ ❬▼❛t❪ ❍✳ ▼❛ts✉♠✉r❛✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✳ ❚r❛♥s❧❛t❡❞ ❢r♦♠ t❤❡ ❏❛♣❛♥❡s❡ ❜② ▼✳ ❘❡✐❞✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ❈❛♠❜r✐❞❣❡ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✲ ✐❝s✱ ✽✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✶✾✽✾✳ ❬▼❛t✶❪ ❍✳ ▼❛ts✉♠✉r❛✱ ❈♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ▼❛t❤❡♠❛t✐❝s ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s✱ ✺✻✳ ❇❡♥❥❛♠✐♥✴❈✉♠♠✐♥❣s P✉❜❧✐s❤✐♥❣ ❈♦✱ ■♥❝✱ ❘❡❛❞✲ ✐♥❣✱ ▼❛ss✱ ✶✾✽✵✳ ❬◆❛❪ ▼✳ ◆❛❣❛t❛✱ ▲♦❝❛❧ r✐♥❣s✳ ❈♦rr❡❝t❡❞ r❡♣r✐♥t✳ ❘♦❜❡rt ❊✳ ❑r✐❡❣❡r P✉❜❧✐s❤✐♥❣ ❈♦✱ ❍✉♥t✐♥❣t♦♥✱ ◆✳ ❨✱ ✶✾✼✺✳ ❬❘♦t❪ ❏✳ ❘♦t♠❛♥✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❍♦♠♦❧♦❣✐❝❛❧ ❆❧❣❡❜r❛✱ ❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛ ✲ ❈❤❛♠♣❛✐❣♥✱ ✷✵✵✵✳