✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❤♦➭♥ t♦➭♥ tr✉♥❣ t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ➤Ị t➭✐ ❦❤➳❝✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ sù ➤å♥❣ ý ❝đ❛ ❝➳ ♥❤➞♥ ✈➭ tỉ ❝❤ø❝✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t➭✐ ❧✐Ư✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❣❤✐ râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ❱ị ❚❤Þ ◆❤✉♥❣ ❳➳❝ ♥❤❐♥ ❝ñ❛ tr➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❳➳❝ ♥❤❐♥ ❝ñ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳❚❙✳ ▲➟ ❚❤Þ ❚❤❛♥❤ ◆❤➭♥ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ s❛✉ ✷ ♥➝♠ ❤ä❝ t❐♣ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✳ ❱í✐ ❧ß♥❣ ❦Ý♥❤ trä♥❣ ✈➭ ❜✐Õt ➡♥ s➞✉ s➽❝ t➠✐ ①✐♥ ➤➢ỵ❝ ❜➭② tá ❧ê✐ ❝➯♠ ➡♥ ❝❤➞♥ t❤➭♥❤ tí✐ P●❙✳❚❙ ▲➟ ❚❤Þ ❚❤❛♥❤ ◆❤➭♥✱ ♥❣➢ê✐ ❝➠ ❦Ý♥❤ ♠Õ♥ ➤➲ ❤Õt ❧ß♥❣ ❣✐ó♣ ➤ì✱ ❞➵② ❜➯♦✱ ➤é♥❣ ✈✐➟♥ ✈➭ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❚❤➳✐ ◆❣✉②➟♥✱ ❧➲♥❤ ➤➵♦ ❦❤♦❛ ❚♦➳♥✱ ❧➲♥❤ ➤➵♦ ❦❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ❝ñ❛ ❚r➢ê♥❣ ➤➲ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❣✐ó♣ ➤ì t➠✐ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐Ư♠ ✈ơ ❤ä❝ t❐♣ ❝đ❛ ♠×♥❤✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➬② ❝➠ ➤➲ t❤❛♠ ❣✐❛ ❣✐➯♥❣ ❞➵② ❝❤♦ ❧í♣ ❈❛♦ ❤ä❝ ❝❤✉②➟♥ ♥❣➭♥❤ ❚♦➳♥ ❦❤♦➳ ✶✾✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝➯♠ ➡♥ ♥❤÷♥❣ ♥❣➢ê✐ t❤➞♥ ②➟✉ tr♦♥❣ ❣✐❛ ➤×♥❤✱ ❜➵♥ ❜❒ ➤➲ ❧✉➠♥ ❝❤♦ t➠✐ ♥✐Ị♠ t✐♥ ✈➭ ➤é♥❣ ❧ù❝ ➤Ĩ ❤ä❝ t❐♣ tèt✳ ✈s❦✐♣ ✵✳✼❝♠ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✐✐✐ ▼ơ❝ ❧ơ❝ ❚r❛♥❣ ▲ê✐ ❝❛♠ ➤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲ê✐ ❝➯♠ ➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▲ê✐ ♥ã✐ ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ♠➠➤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷✳ ❚❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳ ▼➠➤✉♥ ♠ë ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺✳ ❈❤✐Ò✉ ❑r✉❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❈❤➢➡♥❣ ✷✳ ❚Ý♥❤ ❤÷✉ ❤➵♥ ❝ñ❛ t❐♣ Ass ❝ñ❛ ♠ét sè ♠➠➤✉♥ Ext ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳ ❉➲② ❝❤Ý♥❤ q✉② ✈➭ ➤é s➞✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳ ❉➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ Ass ❝ñ❛ ♠ét sè ♠➠➤✉♥ Ext ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸✳ ❙ù tå♥ t➵✐ ❝đ❛ ❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ ✷✳✸✳ ❚Ý♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✱ I ❧➭ ✐➤➟❛♥ ❝đ❛ R ✈➭ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ▼ét ❝➞✉ ❤á✐ ➤➢ỵ❝ q✉❛♥ t➞♠ ❜ë✐ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ♥❤➢ ❉✳ ❘❡❡s ✭✶✾✺✻✮✱ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✻✮ ❦Õt Ass(R/I n ) ♥❣✉②➟♥ Z✱ ❧➭ ❧✐Ö✉ r➺♥❣ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ n ❦❤✐ n ➤đ ❧í♥❄ ❑❤✐ ❝➞✉ tr➯ ❧ê✐ ❧➭ ➤ó♥❣✳ ❈❤ó ý r➺♥❣ ♠ä✐ ✐➤➟❛♥ ❝đ❛ ❝❤Ý♥❤ ✈➭ ♥Õ✉ ♥❣✉②➟♥ I = aZ ✈í✐ a = pα1 pαt t R Z ❧➭ ✈➭♥❤ sè ➤Ò✉ ❧➭ ✐➤➟❛♥ ❧➭ sù ♣❤➞♥ tÝ❝❤ t✐➟✉ ❝❤✉➮♥ ❝ñ❛ sè a t❤× AssZ (Z/I n ) = {p1 Z, , pt Z} ✈í✐ ♠ä✐ n ▼➲✐ ➤Õ♥ ♥➝♠ ✶✾✼✾✱ ▼✳ ❇r♦❞♠❛♥♥ ❬❇❪ ♠í✐ ➤➢❛ r❛ ➤➢ỵ❝ ❝➞✉ tr➯ ❧ê✐ trä♥ ✈Ñ♥ ❝❤♦ ❝➞✉ ❤á✐ ➤ã tr♦♥❣ tr➢ê♥❣ ợ R t ì tr ủ r ữ ú tr ò ú ❝➯ ❝➳❝ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✮✳ ▼✳ ❇r♦❞♠❛♥♥ ❬❇❪ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ t❐♣ ❝ã t❤Ĩ ♣❤ơ t❤✉é❝ ✈➭♦ n, ♥❤➢♥❣ ỉ♥ ➤Þ♥❤ ❦❤✐ n AssR (M/I n M ) ➤đ ❧í♥✱ tø❝ ❧➭ tå♥ t➵✐ sè tù AssR (M/I n M ) = AssR (M/I n0 M ) ✈í✐ ♠ä✐ n ≥ n0 n ❚õ q✉❛♥ s➳t t❤✃② r➺♥❣ M/I M ∼ = TorR (R/I n , M )✱ ▼❡❧❦❡rss♦♥✲ ❙❝❤❡♥✲ ♥❤✐➟♥ n0 s❛♦ ❝❤♦ ③❡❧ ❬▼❙❪ ✶✾✾✸ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ tr➟♥ ❝❤♦ ♠➠➤✉♥ n TorR i (R/I , M ) ✈í✐ ♠ä✐ n i ≥ ❍ä ➤➲ ❝❤Ø r❛ r➺♥❣ AssR TorR i (R/I , M ) ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ n ❦❤✐ n ➤đ ❧í♥ ✈í✐ ♠ä✐ i ≥ ➜å♥❣ t❤ê✐ ❤ä ➤➷t ❝➞✉ ❤á✐ ❧✐Ö✉ AssR ExtiR (R/I n , M ) ❧➭ ➤é❝ ❧❐♣ ✈í✐ n ❦❤✐ n ➤đ ❧í♥❄✳ ❚✉② ♥❤✐➟♥✱ ♥➝♠ ✷✵✵✷✱ ▼✳ ❑❛t③♠❛♥ ❬❑❛t✱ ❍Ư q✉➯ ✶✳✸❪ ➤➲ ①➞② ❞ù♥❣ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❤❛✐ ♣❤➬♥ tư ❤➵♥✳ ❱× AssR t❤á❛ ♠➲♥ I = (x, y)R HI2 (R) ❧➭ t❐♣ ❝♦♥ ❝ñ❛ ❤❛✐ t❐♣ n AssR (R/(x ✈➭ x, y ∈ m n n∈N Ass , y n )R) ♥➟♥ ë tr➢ê♥❣ ❤ỵ♣ ♥➭② n AssR (R/(x n ❦❤➠♥❣ ➤é❝ ❧❐♣ ✈í✐ ❦✐Ư♥ ❝đ❛ ♠➠➤✉♥ , y n )R) n M✱ ❦❤✐ ✈➭ n ✈➭♥❤ Số hóa trung tâm học liệu AssR HI2 (R) ❝❤✐Ò✉ ❧➭ t❐♣ ✈➠ Ext2R (R/I n , M ) n∈N Ass ❧➭ t❐♣ ✈➠ ❤➵♥✱ ✈➭ ❞♦ ➤ã (R, m) ✈➭ Ext2R (R/I n , M ) AssR Ext2R (R/I n , M ) ➤đ ❧í♥✳ ❚õ ➤ã ♥❣➢ê✐ t❛ ➤➷t ✈✃♥ ➤Ị t×♠ ➤✐Ị✉ R✱ ✐➤➟❛♥ I ✈➭ ❞➲② (x1 , , xr ) http://www.lrc-tnu.edu.vn/ ➤Ó ❝➳❝ t❐♣ ✶ AssR ExtiR (R/I n , M ) ✈➭ n∈N ❈❤♦ ❤✐Ö✉ s✳ s ≥ −1 (T )≥s n1 , ,nr AssR (M/(xn1 , , xnr r )M ) ❤÷✉ ❤➵♥✳ ột số ỗ t t ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ T T ❝ñ❛ Spec(R), ❦Ý ❝ã ❝❤✐Ị✉ ❧í♥ ❤➡♥ ❤❛② ❜➺♥❣ ◆➝♠ ✷✵✵✽✱ ▼✳ ❇r♦❞♠❛♥♥ ✈➭ ▲✳ ❚✳ ◆❤❛♥ ❬❇◆❪ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ✈➭ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ (x1 , , xr ) ❧➭ M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s t❤× AssR M/(xn1 , , xnr r )M ❧➭ t❐♣ ❤÷✉ ❤➵♥✳ ≥s n1 , ,nr ❉ï♥❣ ❦Õt q✉➯ ♥➭②✱ ❇r♦❞♠❛♥♥ ✲ ◆❤➭♥ ❬❇◆❪ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❦Õt q✉➯ ❤÷✉ ❤➵♥ ❝❤♦ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ❊①t✳ ❱➱♥ ❞ï♥❣ ❝➠♥❣ ❝ơ ❧➭ M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s✱ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤ ✈➭ ❋✳ ❑❤♦s❤✲❆❤❛♥❣ ❬❑❑❪ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ ♥➭② ❝❤♦ tr➢ê♥❣ ợ ị tứ ỉ ó ữ ❤➵♥ ✐➤➟❛♥ tè✐ ➤➵✐✮✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ♠ét ❝➳❝❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ tr➟♥ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét sè ♠➠➤✉♥ Ext tr♦♥❣ ❜➭✐ ❜➳♦✿ ▼✳ ❇r♦❞♠❛♥♥✱ ▲✳ ❚✳ ◆❤❛♥✱ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ Ext✲♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✻ ✭✷✵✵✽✮✱ ✶✺✷✼✲✶✺✸✻✳ ▲✉❐♥ ✈➝♥ ❝ò♥❣ t❤❛♠ ❦❤➯♦ ♠ét sè ❝❤✐ t✐Õt tr♦♥❣ ❜➭✐ ❜➳♦✿ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤ ❛♥❞ ❋✳ ❑❤♦s❤✲❆❤❛♥❣✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ❝❡rt❛✐♥ s❡ts ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ ❊①t✲♠♦❞✉❧❡s✱ ▼❛♥✉s❝✐♣t❛ ▼❛t❤✳✱ ✶✷✺ ✭✷✵✵✽✮✱ ✸✹✺✲✸✺✷✳ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ❧➭ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✈Ị ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✱ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ♠➠➤✉♥ Ext ✈➭ ❝❤✐Ị✉ ❑r✉❧❧✳ ❈❤➢➡♥❣ ✷ ❧➭ ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥✱ tr×♥❤ ❜➭② ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❬❇◆❪ ✈➭ ❬❑❑❪ ✈Ò tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠ét sè ♠➠➤✉♥ Số hóa trung tâm học liệu Ext✳ http://www.lrc-tnu.edu.vn/ ✷ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❚r♦♥❣ s✉èt ❝❤➢➡♥❣ ♥➭② ❧✉➠♥ ❣✐➯ t❤✐Õt ✐➤➟❛♥ ❝ñ❛ R ✈➭ L ❧➭ ♠ét R✲♠➠➤✉♥ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✱ I ❧➭ ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ❤÷✉ ❤➵♥ s✐♥❤✮✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ t✐Õt ♥➭② ❧➭ ♥❤➽❝ ❧➵✐ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t tr♦♥❣ ❧Ý t❤✉②Õt ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ❈➳❝ ❦Ý ❤✐Ư✉ ✈➭ ❦✐Õ♥ t❤ø❝ ❝đ❛ ❈❤➢➡♥❣ ✶ ➤➢ỵ❝ ✈✐Õt ❞ù❛ t❤❡♦ ❝✉è♥ s➳❝❤ ❝ñ❛ ❍✳ ▼❛ts✉♠✉r❛ ❬▼❛t❪✳ ✶✳✶ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ♠➠➤✉♥ ◆♦❡t❤❡r ◆❤➽❝ ❧➵✐ r➺♥❣ ❝➝♥ ❝đ❛ √ √ ❉Ơ t❤✃② I I ✱ ❦Ý ❤✐Ư✉ ❧➭ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭✐✮ ❚❛ ♥ã✐ ❦Ð♦ t❤❡♦ b∈ ✭✐✐✮ ●✐➯ sö ✈➭ t❛ ♥ã✐ I ❧➭ I I I ợ ị ĩ s I = {a ∈ R | ∃n ∈ N ➤Ó an ∈ I} ❧➭ ♠ét ✐➤➟❛♥ ❝đ❛ √ √ ✈í✐ ♠ä✐ R✳ I ❧➭ ♥❣✉②➟♥ s➡ ♥Õ✉ a, b ∈ R✳ ❧➭ ♥❣✉②➟♥ s➡✳ ➜➷t p= √ ❑❤✐ ➤ã p ✈➭ ab ∈ I, a ∈ /I ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè p✲♥❣✉②➟♥ s➡✳ ✭✐✐✐✮ ▼ét ♣❤➞♥ tÝ❝❤ I = Q1 ∩ ∩ Qn ✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ I I I =R I✳ tr♦♥❣ ➤ã ❧➭ pi ✲♥❣✉②➟♥ s➡✱ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ♥➭② ❝đ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ế ỗ tể ỏ tr tí tr➟♥✮ ✈➭ ❝➳❝ Số hóa trung tâm học liệu Qi pi Qi ❧➭ ❦❤➠♥❣ t❤õ❛ ✭tø❝ ❧➭ ➤➠✐ ♠ét ♣❤➞♥ ❜✐Ưt✳ http://www.lrc-tnu.edu.vn/ ✸ ❈❤ó ý r➺♥❣ ♥Õ✉ √ I ❧➭ ♥❣✉②➟♥ s➡ t❤× I ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ➜✐Ị✉ ợ ú tứ ó ữ ♥❣✉②➟♥ s➡ √ tè✳ ❚✉② ♥❤✐➟♥ ♥Õ✉ I ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ t❤× ✶✳✶✳✷ ❱Ý ❞ơ✳ ❉Ơ t❤✃② tr♦♥❣ ✈➭♥❤ mZ ✈➭ ❝❤Ø ❧➭ ❝➳❝ ✐➤➟❛♥ ❝ã ❞➵♥❣ ■➤➟❛♥ 36Z I I ♠➭ √ I ❧➭ ♥❣✉②➟♥ ❧➭ ♥❣✉②➟♥ s➡✳ Z ❝➳❝ sè ♥❣✉②➟♥✱ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ s➡ ❧➭ ✈í✐ m ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❧➭ ❧➭ ❧ò② t❤õ❛ ❝ñ❛ ♠ét sè ♥❣✉②➟♥ tè✳ 36Z = 4Z ∩ 2Z ∩ 9Z✱ tr♦♥❣ ➤ã 4Z ✈➭ 2Z ❧➭ 2Z✲♥❣✉②➟♥ s➡✱ 9Z ❧➭ 3Z✲♥❣✉②➟♥ s➡✳ ❘â r➭♥❣ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ 2Z tr♦♥❣ ♣❤➞♥ tÝ❝❤ ♥➭② ❧➭ t❤õ❛✳ ❇á t❤➭♥❤ ♣❤➬♥ t❤õ❛ ♥➭② ➤✐ t❛ ➤➢ỵ❝ 36Z = 4Z ∩ 9Z ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝đ❛ 36Z ❈❤ó ý r➺♥❣ ♥Õ✉ ❝ò♥❣ ❧➭ ✐➤➟❛♥ I Q1 , Q2 ❧➭ ❤❛✐ ✐➤➟❛♥ p✲♥❣✉②➟♥ s➡ ❝ñ❛ R Q1 ∩ Q2 tì p s ủ R ì tế từ ỗ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ t❛ ❝ã t❤Ĩ q✉② ♥ã ✈Ị tè✐ t❤✐Ĩ✉ ❜➺♥❣ ❝➳❝❤ ❜á ➤✐ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ t❤õ❛ ✈➭ ❣❤Ð♣ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ã ❝➝♥ ❜➺♥❣ ♥❤❛✉✳ ◆❤×♥ ❝❤✉♥❣ ♠ét ✐➤➟❛♥ ❝ã t❤Ĩ ❦❤➠♥❣ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❤♦➷❝ ❝ã ♥❤✐Ò✉ ❤➡♥ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥✳ ❚✉② ♥❤✐➟♥ t❛ ❝ã tÝ♥❤ ❝❤✃t ❞✉② ♥❤✃t s❛✉ ➤➞②✳ ✶✳✶✳✸ ▼Ư♥❤ ➤Ị✳ ●✐➯ sö I = Q1 ∩ .∩Qn = Q1 ∩ .∩Qm ❧➭ ❤❛✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝ñ❛ I ✱ tr♦♥❣ ➤ã Qi s➡✳ ❑❤✐ ➤ã ●✐➯ sö ➤ã Qi ❧➭ ❧➭ pi ✲♥❣✉②➟♥ s➡ ✈➭ Qi ❧➭ n = m ✈➭ {p1 , , pn } = {p1 , , pn } I = Q1 ∩ ∩ Qn pi ✲♥❣✉②➟♥ ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝đ❛ s➡✳ ❚❤❡♦ ♠Ư♥❤ ➤Ị tr➟♥✱ t❐♣ {p1 , , pn } {p1 , , pn } ❧➭ t❐♣ ❝➳❝ ➢í❝ ♥❣✉②➟♥ tè ❝đ❛ Qi I ì ị t ế I ✱ tr♦♥❣ ❧➭ ①➳❝ ➤Þ♥❤ ❞✉② ♥❤✃t ✭❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝ñ❛ ♥❣✉②➟♥ s➡ pi ✲♥❣✉②➟♥ I ✮✳ ❚❛ ❣ä✐ ❝❤✉♥❣✱ ❝➳❝ t❤➭♥❤ pi tố tể tì Qi ị ❞✉② ♥❤✃t✳ ➜ã ❧➭ tÝ♥❤ ❝❤✃t ❞✉② ♥❤✃t t❤ø ❤❛✐ ❝đ❛ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹ I = Q1 ∩ ∩ Qn = Q1 ∩ ∩ Qn ✶✳✶✳✹ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝ñ❛ I ✱ tr♦♥❣ ➤ã Qi , Qi ❧➭ ❧➭ ❤❛✐ ♣❤➞♥ pi ✲♥❣✉②➟♥ s➡✳ ◆Õ✉ pi tè✐ {p1 , , pn } t❤× Qi = Qi t❤✐Ĩ✉ tr♦♥❣ t❐♣ ❚❤❡♦ ♠Ư♥❤ ➤Ị tr➟♥✱ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ tè ❧✐➟♥ ❦Õt tè✐ t❤✐Ó✉ Qi ø♥❣ ✈í✐ ✐➤➟❛♥ ♥❣✉②➟♥ pi ❧➭ ①➳❝ ➤Þ♥❤ ❞✉② ♥❤✃t✱ t❛ ❣ä✐ ❝❤ó♥❣ ❧➭ ❝➳❝ t❤➭♥❤ ♣❤➬♥ I✳ ♥❣✉②➟♥ s➡ ❝➠ ❧❐♣ ❝đ❛ ❈❤ó♥❣ t❛ ✈õ❛ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ❝➳❝ ✐➤➟❛♥ tr♦♥❣ ♠ét ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✳ ❚r♦♥❣ ♣❤➬♥ t✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ❝♦♥✳ ❚õ ♥❛② ✈Ò s❛✉✱ ❣✐➯ t❤✐Õt R✲♠➠➤✉♥ ✈➭ N ❧➭ ♠ét ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ ❝đ❛ ✶✳✶✳✺ ➜Þ♥❤ ♥❣❤Ü❛✳ ❱í✐ ỗ a (x) = ax é a ♠➠➤✉♥✳ ❚❛ ❣ä✐ tø❝ ❧➭ a ∈ R, a L✳ ϕa : L → L t❛ ❣ä✐ ➳♥❤ ①➵ tr➟♥ L✳ L ❘â r➭♥❣ ϕa ❧➭ ❧ò② ❧✐♥❤ ♥Õ✉ tå♥ t➵✐ sè tù ♥❤✐➟♥ ❝❤♦ ❜ë✐ ❧➭ ♠ét ➤å♥❣ ❝✃✉ n s❛♦ ❝❤♦ ϕna = 0✱ an x = ✈í✐ ♠ä✐ x ∈ L ●✐➯ sư ✈í✐ ♠ä✐ I ❧➭ a∈ / p p✲♥❣✉②➟♥ s➡✳ ❑❤✐ ➤ã ♣❤Ð♣ ♥❤➞♥ ❜ë✐ ✈➭ ❧➭ ❧ị② ❧✐♥❤ ✈í✐ ♠ä✐ a ∈ p✳ a tr➟♥ R/I ❧➭ ➤➡♥ ❝✃✉ ❱× t❤Õ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ư♠ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tõ tr➢ê♥❣ ❤ỵ♣ ✐➤➟❛♥ ❧➟♥ tr➢ê♥❣ ❤ỵ♣ AnnR L = {a ∈ R | aL = 0} ♠➠➤✉♥ ❝♦♥✳ ➜➷t ♠ét ✐➤➟❛♥ ❝đ❛ ✭✐✐✮ ●✐➯ sư ❑❤✐ ➤ã N ❧➭ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡ ❝ñ❛ a tr➟♥ L/N N ❧➭ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡ ❝ñ❛ ✭✐✐✐✮ ▼ét ♣❤➞♥ tÝ❝❤ s➡ ❝ñ❛ L✱ ♥Õ✉ ❧➭ L✳ ➜➷t p = ❧➭ N =L ✈➭ a ∈ R✳ AnnR (L/N )✳ p✲♥❣✉②➟♥ s➡✳ N = Q1 ∩ ∩ Qn ✱ tr♦♥❣ ➤ã Qi ❧➭ ♠➠➤✉♥ ❝♦♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ tÝ❝❤ ♥❣✉②➟♥ s➡ ♥➭② ➤➢ỵ❝ ọ t ọ ế ỗ pi L ❝✃✉ ❤♦➷❝ ❧ị② ❧✐♥❤ ✈í✐ ♠ä✐ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ✈➭ t❛ ♥ã✐ N pi ✲♥❣✉②➟♥ AnnR L R✳ ❚❛ ❣ä✐ AnnR L ❧➭ ❧✐♥❤ ❤ã❛ tư ❝đ❛ L✳ ✶✳✶✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭✐✮ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ ❈❤ó ý r➺♥❣ Qi P❤➞♥ ❧➭ ❦❤➠♥❣ t❤õ❛ ✈➭ ❝➳❝ ❧➭ ➤➠✐ ♠ét ♣❤➞♥ ❜✐Ưt✳ Số hóa trung tâm học liệu N✳ http://www.lrc-tnu.edu.vn/ ✺ ❈❤ó ý r➺♥❣ ♥Õ✉ Q1 , Q2 ❧➭ ❤❛✐ ♠➠➤✉♥ ❝♦♥ p✲♥❣✉②➟♥ s➡ ❝đ❛ L t❤× Q1 ∩Q2 ❝ị♥❣ ❧➭ ♠➠➤✉♥ ❝♦♥ N s➡ ❝đ❛ p✲♥❣✉②➟♥ s➡ ❝đ❛ L ì tế từ ỗ tí t ó t❤Ĩ q✉② ♥ã ✈Ị tè✐ t❤✐Ĩ✉ ❜➺♥❣ ❝➳❝❤ ❜á ➤✐ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ t❤õ❛ ✈➭ ❣❤Ð♣ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ø♥❣ ✈í✐ ❝ï♥❣ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè✳ ❚➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ✐➤➟❛♥✱ ❝❤ó♥❣ t❛ ❝ã ❤❛✐ tÝ♥❤ ❝❤✃t ❞✉② ♥❤✃t s❛✉ ➤➞② ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❝♦♥✳ ✶✳✶✳✼ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư N = Q1 ∩ .∩Qn = Q1 ∩ .∩Qm ❧➭ ❤❛✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝ñ❛ N ✱ tr♦♥❣ ➤ã Qi s➡✳ ❑❤✐ ➤ã ❚❐♣ ❧➭ pi ✲♥❣✉②➟♥ s➡ ✈➭ Qi ❧➭ pi ✲♥❣✉②➟♥ n = m ✈➭ {p1 , , pn } = {p1 , , pn } {p1 , , pn } tr♦♥❣ ♠Ư♥❤ ➤Ị tr➟♥ ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❝➳❝❤ ❝❤ä♥ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝ñ❛ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ó✉ N✳ ❚❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ Qi ø♥❣ ✈í✐ pi tr♦♥❣ t❐♣ {p1 , , pn } ➤➢ỵ❝ ❣ä✐ ❧➭ t❤➭♥❤ ♣❤➬♥ N✳ ♥❣✉②➟♥ s➡ ❝➠ ❧❐♣ ❝đ❛ ✶✳✶✳✽ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư ♥❣✉②➟♥ s➡ ❝➠ ❧❐♣ ❝ñ❛ N N ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ❑❤✐ ➤ã ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❧➭ ①➳❝ ➤Þ♥❤ ❞✉② ♥❤✃t✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♠ä✐ ♠➠➤✉♥ ❝♦♥ ❝đ❛ ♠➠➤✉♥ ◆♦❡t❤❡r ➤Ị✉ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ❚õ ♥❛② ✈Ò s❛✉✱ ❧✉➠♥ ❣✐➯ t❤✐Õt ✈➭ M ❧➭ ❧➭ ✈➭♥❤ ◆♦❡t❤❡r R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❈❤ó ý r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭② M ❧➭ ♠➠➤✉♥ ◆♦❡t❤❡r✳ ❈❤♦ ❚❛ ♥ã✐ N N ❧➭ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ ❧➭ ❜✃t ❦❤➯ q ế ỗ ễ t R N = M ❝ò♥❣ M✳ ✈➭ N ❦❤➠♥❣ ❧➭ ❣✐❛♦ ❝ñ❛ ❤❛✐ a ∈ R✱ ➤➷t N :M a = {m ∈ M | am ∈ N } N :M a ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M ✳ ✶✳✶✳✾ ❇ỉ ➤Ị✳ ◆Õ✉ N ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ♣❤Ð♣ ♥❤➞♥ ❜ë✐ ❜✃t ❦❤➯ q✉② t❤× N N ❧➭ ♥❣✉②➟♥ s➡✳ ❦❤➠♥❣ ♥❣✉②➟♥ s➡✳ ❑❤✐ ➤ã tå♥ t➵✐ a tr➟♥ M/N Số hóa trung tâm học liệu a∈R s❛♦ ❝❤♦ ❦❤➠♥❣ ❧➭ ➤➡♥ ❝✃✉ ✈➭ ❦❤➠♥❣ ❧➭ ❧ị② ❧✐♥❤✳ ❱× http://www.lrc-tnu.edu.vn/ M ✻ ❧➭ ◆♦❡t❤❡r ♥➟♥ ❞➲② t➝♥❣ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❞õ♥❣✱ tø❝ ❧➭ tå♥ t➵✐ ➜➷t N1 = ak M + N ❉♦ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ ❝❤♦ k∈N am ∈ N N2 ⊃ N ✈➭ a ✈➭ s❛♦ ❝❤♦ N :M an = N :M ak N2 = N :M a tr➟♥ M/N N :M a ⊆ N :M a2 ⊆ ❉Ơ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ N2 = N ✳ ak (M/N ) = 0✱ tø❝ ❧➭ ❉♦ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ ak M ⊆ N ❱× t❤Õ a n ≥ k ✈í✐ ♠ä✐ N = N1 ∩ N2 m∈ /N ❦❤➠♥❣ ❧➭ ➤➡♥ ❝✃✉ ♥➟♥ tå♥ t➵✐ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭② t❛ ❝ã m ∈ (N :M a) \ N tr➟♥ M/N N1 ⊃ N ✈➭ ♣❤➯✐ s❛♦ ❉♦ ➤ã ❦❤➠♥❣ ❧ò② ❧✐♥❤ ♥➟♥ N1 = N ❉♦ ➤ã N ❦❤➠♥❣ ❜✃t ❦❤➯ q✉②✱ ✈➠ ❧Ý✳ ✶✳✶✳✶✵ ▼Ư♥❤ ➤Ị✳ ◆Õ✉ M M ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ tì ỗ ủ ề ó tí s➡✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö tå♥ t➵✐ ♠ét ♠➠➤✉♥ ❝♦♥ ❝đ❛ ♥❣✉②➟♥ s➡✳ ❑Ý ❤✐Ư✉ Γ ♥❣✉②➟♥ s➡✳ ❑❤✐ ➤ã Γ = ∅✳ N0 ∈ Γ ♥➟♥ N0 ❉♦ M N0 tr♦♥❣ Γ ♥➟♥ ❦❤➠♥❣ ❝ã ♣❤➞♥ tÝ❝❤ Γ ❦❤➠♥❣ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ❉♦ ➤ã N0 tr♦♥❣ ➤ã N1 , N2 ∈ / Γ✳ N1 = N0 ❉♦ ➤ã N1 , N2 ✈➭ ❝ã ♣❤➬♥ tö ❧➭ ♣❤➬♥ tư ❝ù❝ ➤➵✐ ❝đ❛ N0 M ❝❤ø❛ t❤ù❝ sù N2 = N0 ✳ ❉♦ N0 Γ✳ ❦❤➠♥❣ ❧➭ ❦❤➯ q✉②✱ ♥❣❤Ü❛ ❧➭ tÝ❝❤ ➤➢ỵ❝ t❤➭♥❤ ❣✐❛♦ ❝đ❛ ❤❛✐ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ N0 = N1 ∩ N2 ✱ M ❧➭ ♠➠➤✉♥ ◆♦❡t❤❡r ♥➟♥ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡✳ ❚❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✾✱ t❤Õ ❦❤➠♥❣ ❝ã ♣❤➞♥ tÝ❝❤ ❧➭ t❐♣ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝đ❛ ❝ù❝ ➤➵✐ ✭t❤❡♦ q✉❛♥ ❤Ư ❜❛♦ ❤➭♠✮✳ ●✐➯ sư ❱× M N0 ♣❤➞♥ N0 ✳ ❱× ❧➭ ❝ù❝ ➤➵✐ ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✳ ●✐➯ sö N1 = Q11 ∩ .∩Q1k ✈➭ N2 = Q21 ∩ .∩Q2t ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ N1 ✈➭ N2 t➢➡♥❣ ø♥❣✳ ❑❤✐ ➤ã N0 = N1 ∩N2 = Q11 ∩ .∩Q1k ∩Q21 ∩ .∩Q2t ❧➭ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ✶✳✷ N0 ✱ ➤✐Ị✉ ♥➭② ✈➠ ❧Ý ✈í✐ ❣✐➯ t❤✐Õt N0 ∈ Γ✳ ❚❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❚r♦♥❣ s✉èt ♣❤➬♥ ♥➭② t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt ủ õ r L ỗ L ♠ét R✲♠➠➤✉♥ N ❧➭ ♠ét m ∈ L t❛ ➤➷t AnnR m = {a ∈ R | am = 0}✳ AnnR m ❧➭ ✐➤➟❛♥ ❝đ❛ R✳ Số hóa trung tâm học liệu ✈➭ http://www.lrc-tnu.edu.vn/ ✷✻ i < r ❚✐Õ♣ t❤❡♦✱ ①Ðt ❞➲② ❦❤í♣ → M/(0 :M x) → M → M/xM → ❚❛ ♥❤❐♥ ➤➢ỵ❝ ❞➲② ❦❤í♣ ❞➭✐ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ HIi (M ) → HIi (M/xM ) → HIi+1 (M/(0 :M x)) i ≥ ❚õ ❦Õt q✉➯ dim SuppR (HIi (M/(0 :M x)) ✈í✐ ♠ä✐ ➤➲ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ ✈➭ tõ ❞➲② ❦❤í♣ t❛ ❝ã i < r − ✈í✐ ♠ä✐ tr♦♥❣ I ❧➭ ♠ét dim SuppR (HIi (M/xM )) ❚❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣✱ tå♥ t➵✐ ♠ét ❞➲② s❛♦ ❝❤♦ ♥ã ❧➭ ♠ét x1 , , x r s ✈í✐ ♠ä✐ i < r M/xM ✲❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ò✉ s x2 , , x r > s✳ ❱× ✈❐② M ✲❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ✳ (x1 , , xk ) ✷✳✸✳✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ ❧➭ ♠ét M ✲❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s tr♦♥❣ I ✳ ❚❛ ♥ã✐ (x1 , , xk ) ❧➭ M ✲❞➲② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s I tr♦♥❣ ♥Õ✉ ❦❤➠♥❣ tå♥ t➵✐ ♠ét ♣❤➬♥ tö y∈I s❛♦ ❝❤♦ (x1 , , xk , y) ❧➭ M ✲❞➲② t❤❡♦ ❝❤✐Ò✉ ❧í♥ ❤➡♥ s✳ ❈❤♦ ❝❤Ø ❧➭ s = −1 M ✲❞➲② ✈➭ M = IM ❑❤✐ ➤ã M ✲❞➲② t❤❡♦ ề í q r trờ ợ ỗ M ✲❞➲② s tr♦♥❣ ❧➭ ✈➭ I ➤Ò✉ ♠ë ré♥❣ ➤➢ỵ❝ t❤➭♥❤ ♠ét M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ tr♦♥❣ I ✱ ✈➭ ❝➳❝ M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ tr♦♥❣ I ❝ã ❝❤ó♥❣ ➤é ❞➭✐✱ ➤é ❞➭✐ ❝❤✉♥❣ ♥➭② ❝❤Ý♥❤ ❧➭ depth(I; M ), ➤é s➞✉ ❝ñ❛ M ❑❤✐ I✳ s ≥ 0✱ ❝❤ó♥❣ t❛ ❝ã tÝ♥❤ ❝❤✃t t➢➡♥❣ tù ➤è✐ ✈í✐ ❝➳❝ M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ ✷✳✸✳✹ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư ✭✐✮ ỗ ột tr s dim(M/IM ) > s ❑❤✐ ➤ã M ✲❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ❝ã t❤Ó ♠ë ré♥❣ t❤➭♥❤ M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ✳ ✭✐✐✮ ❈➳❝ M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ ❝❤✉♥❣ ➤é ❞➭✐✱ ✈➭ ➤é ❞➭✐ ❝❤✉♥❣ ♥➭② ❝❤Ý♥❤ ❧➭ sè ♥❣✉②➟♥ ❝❤♦ i I ➤Ò✉ ❝ã ❜Ð ♥❤✃t s❛♦ dim SuppR (HIi (M )) > s✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮✳ ●✐➯ sư tr♦♥❣ I✳ ❱× (x1 , , xr ) dim M/IM ) > s Số hóa trung tâm học liệu ❧➭ ♠ét ♥➟♥ t❛ ❝ã M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ dim M = d > s ≥ http://www.lrc-tnu.edu.vn/ ❱× x1 s ❧➭ ✷✼ M ✲❝❤Ý♥❤ ♠➲♥ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ dim(R/p) > s ❱× ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ s x1 ∈ /p t❤Õ ♥➟♥ x1 ∈ /p ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ p ∈ AssR M p ∈ AssR M d ❉♦ ➤ã t❤á❛ x1 M ✳ ❙✉② r❛ dim(M/x1 M ) = d − ▲➵✐ ❞♦ x1 ∈ I ❧➭ ♥➟♥ d − = dim(M/x1 M ) ≥ dim(M/IM ) > s ❉♦ x2 ❧➭ M/x1 M ✲❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ dim(R/p) > s s x2 ∈ / p ♥➟♥ ❙✉② r❛ x2 ∈ /p ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ p ∈ AssR (M/x1 M ) p ∈ AssR (M/x1 M ) t❤á❛ ♠➲♥ d−1 ❉♦ ➤ã x2 ❧➭ ♣❤➬♥ tö t❤❛♠ sè ❝đ❛ M/x1 M ❈ø t✐Õ♣ tơ❝ ❧❐♣ ❧✉❐♥ ♥❤➢ tr➟♥ t❛ s✉② r❛ xi ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ M/(x1 , , xi−1 )M ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ t❤➭♥❤ ♠ét M ✲❞➲② s t❤❡♦ ❝❤✐Ò✉ >s (x1 , , xr ) > s ề ứ tỏ ỗ ó tể é ❞➭✐ ✭s❛✉ ❦❤➠♥❣ q✉➳ s tr♦♥❣ tr♦♥❣ I✳ ✈➭ I ❞♦ ➤ã I✳ d ❜➢í❝✮ ➤Ĩ ➤Ị✉ ❝ã t❤Ĩ ♠ë ré♥❣ t❤➭♥❤ ♠ét M ✲❞➲② > s tr♦♥❣ I (y1 , , yt ) ❧➭ ❤❛✐ ❚❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ M ✲❞➲② r = t dim(SuppR HIi (M )) dim(SuppR HIi (M )) s s ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ i ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ ●✐➯ t❤✐Õt r➺♥❣ r < t i < t r dim(SuppR (HIi (M/(x1 , , xk )M ))) dim(HI0 (M/(x1 , , xr )M )) ▼Ư♥❤ ➤Ị ✷✳✸✳✷ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❧➭ M/(x1 , , xr )M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ ❞➲② s ❈❤ó ý r➺♥❣ I✳ r < ❚➢➡♥❣ tù ♥❤➢ tr♦♥❣ k r➺♥❣ r−k ✈➭ ♠ä✐ k r s ❉♦ ➤ã t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ r = 1✱ tå♥ t➵✐ ♠ét ♣❤➬♥ tư y ∈ I t❤❡♦ ❝❤✐Ị✉ tr♦♥❣ s ✈í✐ ♠ä✐ i r = ❚❤❡♦ ▼Ư♥❤ ➤Ị ❝❤ø♥❣ ♠✐♥❤ ▼Ư♥❤ ➤Ị ✷✳✸✳✷ t❛ ❝ã t❤Ĩ s✉② r❛ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ ❱× ✈❐② M ✲❞➲② M ✲❞➲② tr♦♥❣ ❦❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt ✷✳✸✳✷ t ó ỗ tố t ề ❤➡♥ ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ò✉ ✭✐✐✮✳ ❈❤♦ d = dim M I tr♦♥❣ ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ò✉ t✱ M ì tế ỗ M t ề ❧í♥ ❤➡♥ s tr♦♥❣ ❝ã ➤é ❞➭✐ ❦❤➠♥❣ ✈➢ỵt q✉➳ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ t✱ i = 1, , r dim(M/IM ) > s tì ỗ M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s tr♦♥❣ I ✈❐②✱ ❦❤✐ I ✈í✐ ♠ä✐ > s✱ tø❝ ❧➭ (x1 , , xr , y) s❛♦ ❝❤♦ ❧➭ y M ✲❞➲② ➜✐Ị✉ ♥➭② ❧➭ ♠➞✉ t❤✉➱♥ ✈í✐ tÝ♥❤ tè✐ ➤➵✐ ❝ñ❛ (x1 , , xr ) tr♦♥❣ I ✳ ❱❐②✱ t✃t ❝➯ ❝➳❝ M ✲❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ➤Ò✉ ❝ã ❝❤✉♥❣ ➤é ❞➭✐✱ ✈➭ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✸✳✷✱ ➤é ❞➭✐ ❝❤✉♥❣ ♥➭② Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✷✽ ❝❤Ý♥❤ ❧➭ ❝❤Ø sè i ❜Ð ♥❤✃t s❛♦ ❝❤♦ dim(SuppR (HIi (M ))) > s✳ ❉➢í✐ ➤➞② ❝❤ó♥❣ t❛ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ dim M/IM ) s t❤× ❦❤➠♥❣ tå♥ t➵✐ M ✲❞➲② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s tr♦♥❣ I ✳ ✷✳✸✳✺ ❇ỉ ➤Ị✳ ◆Õ✉ t➵✐ ♠ét dim(M/IM ) s tì ỗ số n > ❧✉➠♥ tå♥ M ✲❞➲② t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ❦❤➠♥❣ tå♥ t➵✐ ✈í✐ ♠ä✐ n✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✱ M ✲❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ t❤❡♦ ❝❤✐Ò✉ > s tr♦♥❣ I ❈❤ø♥❣ ♠✐♥❤✳ ❉♦ t❤á❛ ♠➲♥ ❝ã ➤é ❞➭✐ dim(M/IM ) dim(R/p) > s p ∈ (AssR M )>s s ♥➟♥ I ⊂ p ❉♦ ➤ã t❛ ❝❤ä♥ ➤➢ỵ❝ ▲➵✐ ❞♦ ✈í✐ ♠ä✐ p ∈ AssR M x1 ∈ I s❛♦ ❝❤♦ x1 ∈ / p s I ⊂p ✈í✐ ♠ä✐ dim(M/IM ) ♥➟♥ p ∈ AssR (M/x1 M ) t❤á❛ ♠➲♥ dim(R/p) > s ❉♦ ➤ã t❛ ❝❤ä♥ ➤➢ỵ❝ x2 ∈ I s❛♦ ❝❤♦ x2 ∈ / p ✈í✐ ♠ä✐ tr➟♥✱ t❛ ❝❤ä♥ ➤➢ỵ❝ ❞➲② p ∈ (AssR M/x1 M )>s (x1 , , xn ) AssR M/(x1 , , xi−1 )M ❧í♥ ❤➡♥ >s tr♦♥❣ ❱× t❤Õ I s❛♦ ❝❤♦ (x1 , , xn ) xi ∈ / p ❧➭ ✈í✐ ♠ä✐ M ✲❞➲② p∈ t❤❡♦ ❝❤✐Ò✉ s tr♦♥❣ I ✱ ❞➲② ♥➭② râ r➭♥❣ ❝ã ➤é ❞➭✐ n✳ P❤➬♥ t✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ s✳ ✷✳✸✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ▼ét ❞➲② ♠ét ❈ø t✐Õ♣ tơ❝ q✉➳ tr×♥❤ (x1 , , xn ) ❝➳❝ ♣❤➬♥ tư tr♦♥❣ m ➤➢ỵ❝ ❣ä✐ ❧➭ M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ♥Õ✉ ♠ä✐ ❤♦➳♥ ✈Þ ❝đ❛ ❞➲② ♥➭② ➤Ị✉ ❧➭ M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s✳ ❚r➢í❝ ❦❤✐ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ❝đ❛ ❞➲② ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ q✉②✳ ❑Ý ❤✐Ư✉ s✱ ❝❤ó♥❣ t❛ q✉❛♥ t➞♠ ➤Õ♥ ❦❤➳✐ ♥✐Ö♠ ❞➲② I ✲❧ä❝ ❝❤Ý♥❤ I ➤➢ỵ❝ Var(I) ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤ø❛ I ✳ ✷✳✸✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ❬◆❙❪ ▼ét ❞➲② ❣ä✐ ❧➭ ♠ét ❞➲② I ✲❧ä❝ (x1 , , xr ) ❝❤Ý♥❤ q✉② ➤è✐ ✈í✐ ❞➲② ❝❤Ý♥❤ q✉② ♥❣❤❒♦ ✈í✐ ♠ä✐ Số hóa trung tâm học liệu M ❝➳❝ ♣❤➬♥ tö tr♦♥❣ ♥Õ✉ (x1 , , xr ) p ∈ Spec(R) \ Var(I) ❧➭ ♠ét Mp ✲ ❉Ô t❤✃② r➺♥❣ ❞➲② http://www.lrc-tnu.edu.vn/ ✷✾ (x1 , , xr ) tr♦♥❣ I ♠ä✐ ❧➭ ♠ét ❞➲② I ✲❧ä❝ ❝❤Ý♥❤ q✉② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ xi ∈ / p ✈í✐ p ∈ AssR (M/(x1 , , xi−1 )M ) \ Var(I) ✈➭ ♠ä✐ i = 1, , r ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❞➲② I ✲❧ä❝ ❝❤Ý♥❤ q✉② ✈➭ t❤❡♦ ➜Þ♥❤ í tr tố n t t r ỗ sè tù ♥❤✐➟♥ n ➤é ❞➭✐ ❱× t❤Õ ❦❤➠♥❣ tå♥ t➵✐ ❞➲② ✷✳✸✳✽ ❇ỉ ➤Ị✳ ✭❝❢✳ ❬◆❙✱ ✸✳✹❪✮✳ ◆Õ✉ M I ✲❧ä❝ I ✲❧ä❝ ❝❤Ý♥❤ q✉② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐✳ ❉➢í✐ ➤➞② ❧➭ I ✲❧ä❝ ❝❤Ý♥❤ q✉② ❤❛② ➤➢ỵ❝ sư ❞ơ♥❣✳ ♠ét tÝ♥❤ ❝❤✃t ❝đ❛ ❞➲② ➤è✐ ✈í✐ ❧✉➠♥ tå♥ t➵✐ ♠ét ❞➲② (x1 , , xr ) ❧➭ ♠ét ❞➲② I ✲❧ä❝ ❝❤Ý♥❤ q✉② t❤× HIj (M ) j H(x (M ), , ,xr )R = ♥Õ✉ j s ❝ã ➤é ❞➭✐ r ➤å♥❣ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ M t❤ê✐ ❧➭ ✭✐✐✮ M ✲❞➲② s, r ∈ N✱ r ≥ 1✱ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✳ dim(Supp(HIi (M ))) s ✈í✐ ♠ä✐ i < r✳ ❈❤ø♥❣ ♠✐♥❤✳ ❑❤➻♥❣ ➤Þ♥❤ ✭✐✮⇒✭✐✐✮ s✉② r❛ ♥❣❛② tõ ▼Ư♥❤ ➤Ị ✷✳✸✳✷ ✭✐✮⇒✭✐✐✮✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❦❤➻♥❣ ➤Þ♥❤ ✭✐✐✮⇒✭✐✮ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ C1 := AssR M ❱× dim(HI0 (M )) p ∈ AssR M ≥s+1 s ≥s+1 r ❈❤♦ r = ➜➷t ∪ AssR M \ V (I) t❤❡♦ ❣✐➯ t❤✐Õt ✭✐✐✮ ♥➟♥ t❛ s✉② r❛ I ⊆ p ọ ì tế t ị í tr ♥❣✉②➟♥ tè✱ tå♥ t➵✐ ♠ét ♣❤➬♥ tö x1 ∈ I M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ s❛♦ ❝❤♦ x1 ∈ / p ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ ø♥❣ ✈í✐ ♠➠➤✉♥ Số hóa trung tâm học liệu ✈í✐ ♠ä✐ p ∈ C1 > s ❘â r➭♥❣ r➺♥❣ ✈➭ ❧➭ ♠ét I ✲❞➲② x1 ❧➭ ♠ét ❧ä❝ ❝❤Ý♥❤ q✉② M, ❦Õt q✉➯ ➤ó♥❣ ✈í✐ r = http://www.lrc-tnu.edu.vn/ ✸✵ ❈❤♦ r > ✈➭ ❣✐➯ t❤✐Õt r➺♥❣ ♠Ư♥❤ ➤Ị ➤➲ ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ r − ❑❤✐ ➤ã tå♥ t➵✐ ❞➲② (x1 , , xr−1 ) tr♦♥❣ I ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ M✳ t❐♣ ❤ỵ♣ {1, , r −1}, ❞➲② (xj )j∈J ❚❤❡♦ ▼Ư♥❤ ➤Ị t tết ỗ t í q✉② t❤❡♦ ❝❤✐Ò✉ tr♦♥❣ M ✲❞➲② ❦❤➠♥❣ ➤✐Ò✉ > s ✈➭ ✈õ❛ ❧➭ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤✐Ò✉ ❦✐Ư♥ ø♥❣ ✈í✐ ♠➠➤✉♥ ❝đ❛ t❐♣ s❛♦ ❝❤♦ ♥ã ✈õ❛ ❧➭ >s {1, , r − 1}✱ tr♦♥❣ I ❝ã t❤Ĩ ➤➢ỵ❝ ♠ë ré♥❣ t❤➭♥❤ ♠ét tå♥ t➵✐ ♠ét (M/ j∈J I ❚õ ➤ã t❛ s✉② r❛ I ⊆ p ✈í✐ ♠ä✐ p ∈ ✈➭ ♠ä✐ t❐♣ ❝♦♥ J ❝ñ❛ t❐♣ {1, , r − 1} t❛ ❝ã t❤Ĩ ❝❤ä♥ ➤➢ỵ❝ ♠ét ♣❤➬♥ tư r✳ ✈í✐ ➤é ❞➭✐ xr ∈ I J ❝đ❛ M ì tế ỗ t xj M ) AssR M/ t❤❡♦ ❝❤✐Ò✉ j∈J J >s xj M ≥s+1 ❚❤❡♦ ➜Þ♥❤ ❧Ý tr➳♥❤ ♥❣✉②➟♥ tè✱ s❛♦ ❝❤♦ xr ∈ /p ✈í✐ ♠ä✐ p ∈ Cr , tr♦♥❣ ➤ã Cr : = AssR (M/ xj M ) j∈J J⊆{1, ,r−1} xj M ) \ V (I) AssR (M/ j∈J J⊆{1, ,r−1} (x1 , , xr ) ❧➭ ♠ét M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❚r➢í❝ ❤Õt✱ t❛ ❝❤Ø r❛ r➺♥❣ ❝❤✐Ò✉ > s ❈❤♦ σ (xσ(1) , , xσ(r) ) ❧➭ ♠ét ❤♦➳♥ ✈Þ ❝đ❛ ❦❤➠♥❣ ❧➭ M ✲❞➲② sè ♥❣✉②➟♥ ❜Ð ♥❤✃t s❛♦ ❝❤♦ t❤❡♦ ❝❤✐Ị✉ ❱× t❤Õ p∈ t❤❡♦ ❝❤✐Ị✉ (xσ(1) , , xσ(n) ) ●✐➯ sö ♣❤➯♥ ❝❤ø♥❣ r➺♥❣ > s ●ä✐ ❦❤➠♥❣ ❧➭ n ∈ {1, , r} M ✲❞➲② ❧➭ ❝❤Ý♥❤ q✉② ✈➭ (xσ(1) , , xσ(n) ) s❛♦ ❝❤♦ ≥s+1 ❦❤➠♥❣ ❧➭ Mp ✲ (xσ(1) , , xσ(i−1) , xσ(i+1) , , xσ(n) , xσ(i) ) ❦❤➠♥❣ ❧➭ ❝❤Ý♥❤ q✉②✳ ➜➷t ♥➟♥ t❛ t❤✃② r➺♥❣ AssR (M/(xσ(1) , , xσ(n−1) )M ) xσ(1) , , xσ(n) ∈ p ❞➲② ❝❤Ý♥❤ q✉②✳ ❙✉② r❛ Mp ✲❞➲② {1, , r} > s ❑❤✐ ➤ã r = σ(i) ✈í✐ i < n t❤❡♦ ❝➳❝❤ ❝❤ä♥ ❝đ❛ xr , ✈➭ tå♥ t➵✐ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè xσ(n) ∈ p ≥s+1 J := {j ∈ N | j (xσ(j) )j∈J ❦❤➠♥❣ ❧➭ ❞➲② ❝❤Ý♥❤ q✉② ❝ñ❛ ❧➭ ♠ét n, j = i} Mp ✲❞➲② ❱× dim(R/p) > s ❝❤Ý♥❤ q✉②✳ ❉♦ ➤ã xσ(j) Mp ✳ ❱× t❤Õ✱ tå♥ t➵✐ q ∈ Spec(R) Mp / j∈J Số hóa trung tâm học liệu xr = xσ(i) http://www.lrc-tnu.edu.vn/ ✸✶ ✈í✐ q ⊆ p s❛♦ ❝❤♦ xr ∈ qRp ∈ AssRp Mp / xr ∈ q ∈ AssR (M/ j∈J xσ(j) M ) xσ(j) Mp ❚õ ➤➞② t❛ ❝ã ≥s+1 j∈J ⊆ Cr ➜✐Ò✉ ♥➭② ❧➭ ✈➠ ❧Ý✳ ❈✉è✐ ❝ï♥❣ t❛ ❝➬♥ ❝❤Ø r❛ ➤✐Ị✉ ❦✐Ư♥ ø♥❣ ✈í✐ ♠➠➤✉♥ (x1 , , xr ) ❧➭ ♠ét I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ M✳ ●✐➯ t❤✐Õt r➺♥❣ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ø♥❣ ✈í✐ ♠➠➤✉♥ M (xσ(1) , , xσ(r) ) σ ✈í✐ ♠ét ❤♦➳♥ ✈Þ ❦❤➠♥❣ ❧➭ ♥➭♦ ➤ã ❝đ❛ t❐♣ {1, , r} ●ä✐ n ❧➭ sè ♥❣✉②➟♥ ❜Ð ♥❤✃t s❛♦ ❝❤♦ (xσ(1) , , xσ(n) ) ❦❤➠♥❣ ❧➭ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ø♥❣ ✈í✐ ♠➠➤✉♥ M✳ ❚❤❡♦ ❝➳❝❤ ❝❤ä♥ xr , t❛ t❤✃② r➺♥❣ r = σ(i) ✈í✐ ♠ét sè i < n ♥➭♦ ➤ã✱ ✈➭ tå♥ t➵✐ p ∈ AssR (M/(xσ(1) , , xσ(n−1) )M ) \ V (I) s❛♦ ❝❤♦ ❧➭ xσ(n) ∈ p✳ Mp ✲❞➲② ❦❤➠♥❣ ❧➭ ❱× t❤Õ xσ(1) , , xσ(n) ∈ p ❝❤Ý♥❤ q✉②✳ ❙✉② r❛ Mp ✲❞➲② ✈➭ (xσ(1) , , xσ(n) ) ❦❤➠♥❣ (xσ(1) , , xσ(i−1) , xσ(i+1) , , xσ(n) , xσ(i) ) ❝❤Ý♥❤ q✉②✳ ➜➷t J := {j ∈ N : j p ∈ / V (I) ♥➟♥ t❛ ❜✐Õt r➺♥❣ xr = xσ(i) ❦❤➠♥❣ ❧➭ ♣❤➬♥ tö ❝❤Ý♥❤ q✉② ø♥❣ ✈í✐ ♠➠➤✉♥ (xσ(j) )j∈J ❧➭ ♠ét Mp ✲❞➲② n, j = i} ❱× ❝❤Ý♥❤ q✉②✳ ❉♦ ➤ã Mp / xσ(j) Mp ✳ j∈J ❉♦ ➤ã tå♥ t➵✐ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè q ∈ Spec(R) ✈í✐ q ⊆ p s❛♦ ❝❤♦ xr ∈ qRp ∈ AssRp Mp / xσ(j) Mp j∈J ❙✉② r❛ xr ∈ q ∈ AssR (M/ xσ(j) M ) j∈J ≥s+1 \ V (I) ⊆ Cr , ➤✐Ò✉ ♥➭② ❧➭ ✈➠ ❧Ý✳ P❤➬♥ ❝✉è✐ ❝đ❛ t✐Õt ♥➭② ❞➭♥❤ ➤Ĩ tr×♥❤ ❜➭② ♠ét ❦Õt q✉➯ ❤÷✉ ❤➵♥ ✈Ị t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ø♥❣ ✈í✐ ❤➡♥ M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ ❧í♥ s✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✷ ✷✳✸✳✶✶ ▼Ư♥❤ ➤Ị✳ ❈❤♦ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ s ∈ N ❈❤♦ (x1 , , xr ) ❧➭ ♠ét M ✲❞➲② ❦❤➠♥❣ ➤✐Ò✉ > s ❑❤✐ ➤ã t❛ ❝ã AssR M/(xn1 , , xnr r )M ≥s n1 , ,nr ∈N = AssR (M/(x1 , , xr )M )) AssR M/(xn1 , , xnr r )M ➜➷❝ ❜✐Öt✱ n1 , ,nr ∈N ≥s ❧➭ t❐♣ ❤÷✉ ❤➵♥✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ♠Ư♥❤ ➤Ò ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ t❛ ✈✐Õt r ❈❤♦ r = ✈➭ x1 = x✳ ❈❤ó♥❣ t❛ sÏ ❝❤Ø r❛ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ n1 = n r➺♥❣ AssR (M/xn M ) ❚r➢ê♥❣ ❤ỵ♣ n = ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ≥s ❧➭ râ r➭♥❣✳ ❈❤♦ n − AssRp (Mp /xn Mp ) ✈× t❤Õ ≥s ◆Õ✉ ❈❤♦ p ∈ ⊆ AssR (M/xM ) n > AssR (M/xn M ) dim(R/p) > s pRp ∈ AssRp (Mp /xMp ) ✈➭ ❣✐➯ t❤✐Õt ❦Õt q✉➯ ➤➲ ➤ó♥❣ ❙✉② r❛ t❤× x ❑❤✐ ➤ã pRp ∈ ≥s ❧➭ ♣❤➬♥ tö Mp ✲❝❤Ý♥❤ q✉② ✈➭ p ∈ AssR (M/xM ) ●✐➯ t❤✐Õt r➺♥❣ dim(R/p) = s ❚õ ❞➲② ❦❤í♣ → xn−1 M/xn M → M/xn M → M/xn−1 M → 0, t❛ ❝ã p ∈ AssR (M/xn−1 M ) ∪ AssR (xn−1 M/xn M ) ❚r➢í❝ ❤Õt t❛ ❣✐➯ t❤✐Õt r➺♥❣ p ∈ AssR (M/xn−1 M )✳ p ∈ AssR (M/xM )✳ ❑❤✐ ➤ã t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❛ ♣❤➯✐ ❝ã ❱× t❤Õ t❛ ❣✐➯ t❤✐Õt p ∈ AssR (xn−1 M/xn M ) ❳Ðt ❞➲② ❦❤í♣ → (xn M :M xn−1 )/xM → M/xM → xn−1 M/xn M → ➜➷t K = (xn M :M xn−1 )/xM ❚r➢í❝ ❤Õt ❣✐➯ t❤✐Õt r➺♥❣ p∈ / SuppR (K)✳ ❑❤✐ ➤ã ❞➲② ❦❤í♣ tr➟♥ ❝❤Ø r❛ r➺♥❣ M/xM Số hóa trung tâm học liệu p ∼ = xn−1 M/xn M p http://www.lrc-tnu.edu.vn/ ✸✸ ❱× p ∈ AssR (xn−1 M/xn M ) ♥➟♥ t❛ ❝ã p ∈ AssR (M/xM ) ❱× t❤Õ t❛ ❣✐➯ t❤✐Õt p ∈ SuppR (K) ❱× x ❧➭ ♣❤➬♥ tư Mq ✲❝❤Ý♥❤ q✉② ♥❣❤❒♦ ♥➟♥ Kq = ✈í✐ ♠ä✐ q ∈ SuppR (M ) r❛ p t❤♦➯ ♠➲♥ dim(R/q) > s✳ ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ tr♦♥❣ t❐♣ SuppR (K), ❉♦ ➤ã dim(K) ✈➭ ❞♦ ➤ã s ❙✉② p ∈ AssR (K) ⊆ AssR (M/xM ) ❱❐② ♠Ư♥❤ ➤Ị ❧➭ ➤ó♥❣ ✈í✐ r = ❈❤♦ ❈❤♦ r >1 ✈➭ ❣✐➯ t❤✐Õt r➺♥❣ ♠Ư♥❤ ➤Ị ➤➲ ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ n1 , , n r t❤❡♦ ❝❤✐Ị✉ tr➢ê♥❣ ❤ỵ♣ ❧➭ ❝➳❝ sè ♥❣✉②➟♥ ❞➢➡♥❣ t✉ú ý✳ ❱× n r−1 > s ➤è✐ ✈í✐ ♠➠➤✉♥ M/(xn1 , , xr−1 )M ❧➭ ♣❤➬♥ tö ❝❤Ý♥❤ q✉② ♥➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ tõ r = tÝ♥❤ ❝❤✃t s❛✉ AssR M/(xn1 , , xnr r )M ❱× xr r − x1 , , xr−1 ❧➭ ♠ét n ≥s r−1 ⊆ AssR M/(xn1 , , xr−1 , xr )M M/xr M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ >s ♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❛ s✉② r❛ n r−1 AssR M/(xn1 , , xr−1 , xr )M ≥s n r−1 AssR M/(xr , xn1 , , xr−1 )M ≥s = ⊆ AssR M/(x1 , , xr )M ❱❐② ♠Ư♥❤ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ❤♦➭♥ t♦➭♥✳ ◆❤×♥ ❝❤✉♥❣✱ ❦❤✐ ♥❤✃t t❤✐Õt ❧➭ s ≥ 0✱ ❤♦➳♥ ✈Þ ❝đ❛ ♠ét M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ > s ❦❤➠♥❣ M ✲❞➲② t❤❡♦ ❝❤✐Ò✉ K[[x1 , x2 , x3 , x4 , x5 ]] sè tr➟♥ ♠ét tr➢ê♥❣ tè✐ ➤➵✐ ❞✉② ♥❤✃t ❑❤✐ ➤ã ➤➢ỵ❝ ❧➭ M ❧➭ (x5 , x2 ) > s✳ ❈❤➻♥❣ ❤➵♥✱ ❝❤ä♥ ❧➭ ✈➭♥❤ ❝➳❝ ỗ ỹ từ ì tứ K ó R R ❈❤ä♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ M ✲❞➲② ❈❤ä♥ R= ❜✐Õ♥ ✈í✐ ❤Ư ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ m = (x1 , , x5 )R✳ ❧➭ ♠ét s = M = R/(x1 ) ∩ (x2 , x3 , x4 ) dim M = 4✳ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ 1✱ ❚❛ ❝ã t❤Ĩ ❦✐Ĩ♠ tr❛ ♥❤➢♥❣ (x2 , x5 ) ❦❤➠♥❣ ❧➭ ♠ét M ✲❞➲② M ✲❞➲② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ 1✳ ❚õ ▼Ư♥❤ ➤Ị ✷✳✸✳✶✶ t❛ ❝ã ♥❣❛② ❦Õt q✉➯ s❛✉ ➤➞②✳ ✷✳✸✳✶✷ ❍Ö q✉➯✳ ❈❤♦ s ∈ {−1, 0, 1} Số hóa trung tâm học liệu ●✐➯ sư (x1 , , xr ) http://www.lrc-tnu.edu.vn/ ✸✹ ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s✳ ❑❤✐ ➤ã t❐♣ ❤ỵ♣ AssR M/(xn1 , , xnr r )M \ {m} n1 , ,nr ∈N ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ AssR M/(xn1 , , xnr r )M n1 , , nr ✈➭ ❧➭ n1 , ,nr ∈N t❐♣ ❤÷✉ ❤➵♥✳ ✷✳✹ Ass ❝đ❛ ♠➠➤✉♥ Ext ❚Ý♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ❚r♦♥❣ t✐Õt ♥➭②✱ ❧✉➠♥ ❣✐➯ t❤✐Õt (R, m) ✈➭ M ❧➭ I ❧➭ ✐➤➟❛♥ ủ tr ị R ữ s dim M = d ❚r➢í❝ ❤Õt✱ ❝❤ó♥❣ t❛ ♥❤➽❝ ❧➵✐ ♠ét ❦Õt q✉➯ q✉❛♥ trä♥❣ ✈Ò t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ✷✳✹✳✶ ❇ỉ ➤Ị✳ ❈❤♦ Ext t➵✐ ❝✃♣ r = depth(I, M ) I ⊆ R ❧➭ ✐➤➟❛♥ ✈í✐ depth(I, M ) = r ❑❤✐ ➤ã t❛ ❝ã AssR (ExtrR (R/I, M )) = AssR (HIr (M )) r−1 ✷✳✹✳✷ ❇ỉ ➤Ị✳ ❈❤♦ SuppR ExtiR (R/I, M ) ❑❤✐ ➤ã r ∈ N ➜➷t Pr = i=0 AssR ExttR (R/I n , M ) ∪ Pr = AssR ExtrR (R/(an1 , , ank k ), M ) ∪ Pr = AssR ExtrR (R/I, M ) ∪ Pr = AssR HIr (M ) ∪ Pr ✈í✐ ♠ä✐ ❤Ư s✐♥❤ (a1 , , ak ) ❝ñ❛ I ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ p ∈ SuppR M ✈➭ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ p∈ / Pr ❱í✐ ♠ä✐ i < r ExtiRp (Rp /IRp , Mp ) ∼ = ExtiR (R/I, M ) ❱× t❤Õ n, n1 , , nk p t❛ ❝ã = depth(IRp , Mp ) ≥ r ❈❤♦ n, n1 , , nk ∈ N ❘â r➭♥❣ depth(I n Rp , Mp ) = depth(IRp , Mp ) = depth((an1 , , ank k )Rp , Mp ) Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✺ ❚r➢í❝ ❤Õt t❛ ❣✐➯ t❤✐Õt r➺♥❣ r ♠➠➤✉♥ ExtR p ✈➭ depth(IRp , Mp ) > r✳ ❑❤✐ ➤ã t✃t ❝➯ ❝➳❝ Rp ✲ r (Rp /I n Rp , Mp )✱ ExtrRp (Rp /(an1 , , ank k )Rp , Mp )✱ HIR (Mp ) p ExtrRp (Rp /IRp , Mp ) ♥➭♦ tr♦♥❣ sè t❐♣ AssR ➤Ị✉ ❜➺♥❣ 0✳ ❱× t❤Õ p ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ❜✃t ❦× t❐♣ ExtrR (R/I n , M ), AssR ExtrR (R/(an1 , , ank k ), M )✱ AssR ExtrR (R/I, M ) ✈➭ AssR HIr (M ) ●✐➯ sö depth(IRp , Mp ) = r ❱× rad(I) = rad(I n ) = rad((an1 , , ank k )R) ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✶ t❛ ❝ã AssRp ExtrRp (Rp /I n Rp , Mp ) = AssRp HIrn Rp (Mp ) r = AssRp HIR (Mp ) p = AssRp ExtrRp (Rp /IRp , Mp ) , r AssRp ExtrRp (Rp /(an1 , , ank k )Rp , Mp ) = AssRp HIR (Mp ) p ❙✉② r❛ p ∈ AssR (ExtrR (R/I n , M ) ⇔ p ∈ AssR (HIr (M )) ⇔ p ∈ AssR (ExtrR (R/I, M ) ⇔ p ∈ AssR (ExtrR (R/(an1 , , ank k ), M ) ❚õ ➤➞② t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❇ỉ ➤Ị s❛✉ ➤➞② s✉② r❛ ♥❣❛② ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ r ✈➭ sư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✹✳✶ ổ ề ổ ề ỗ số ♥❣✉②➟♥ r ≥ t❛ ❝ã r r SuppR ExtiR (R/I n , M ) SuppR ExtiR (R/I, M ) = i=0 n∈N i=0 r SuppR HIi (M ) = i=0 Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸✻ ❈❤ó ý r➺♥❣ ➤➻♥❣ t❤ø❝ tr♦♥❣ ❇ỉ ➤Ị ✷✳✹✳✷ s❛✉ ➤➞② AssR ExtrR (R/I, M ) ∪ Pr = AssR HIr (M ) ∪ Pr ✈➭ ➤➻♥❣ t❤ø❝ tr♦♥❣ ❇æ ➤Ò ✷✳✹✳✸ s❛✉ ➤➞② r r SuppR ExtiR (R/I, M ) SuppR HIi (M ) = i=0 i=0 ➤➲ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ♠ét ❜➭✐ ❜➳♦ ❝ñ❛ ◆✳ ❚✳ ❈➢ê♥❣ ✈➭ ◆✳ ❱✳ ❍♦➭♥❣ ♥➝♠ ✷✵✵✺✳ ✷✳✹✳✹ ❑Ý ❤✐Ư✉✳ ❱í✐ ỗ số tự ệ i ỗ I ủ R ỗ a = (a1 , , ak ) ❝➳❝ ♣❤➬♥ tư ❝đ❛ R t❛ ➤➷t AssR ExtiR (R/I n , M ); T i (I, M ) : = n∈N AssR ExtiR (R/(an1 , , ank k ), M ) T i (a, M ) : = n1 , ,nk ∈N ➜Þ♥❤ ❧Ý s❛✉ ➤➞② ❧➭ ♠ét tr♦♥❣ ❤❛✐ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭②✳ ✷✳✹✳✺ ➜Þ♥❤ ❧ý✳ ❈❤♦ i < t✳ ✈í✐ ♠ä✐ ♥❣✉②➟♥ s, t ∈ N ✈í✐ t ≥ 1✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ ❤Ư s✐♥❤ t✱ ❝➳❝ t❐♣ ❤ỵ♣ T r (I, M ) r r ●✐➯ sö dim(SuppR HIi (M )) a = (a1 , , ak ) ≥s ✈➭ T r (a, M ) ❝ñ❛ ≥s I s ✈➭ ♠ä✐ sè ❝❤ø❛ tr♦♥❣ t❐♣ AssR ExtiR (R/I, M )✳ ❤÷✉ ❤➵♥ i=0 ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ r t ❧➭ ♠ét sè ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠✳ ➜➷t r−1 SuppR ExtiR (R/I, M ) Pr = i=0 r−1 ❚❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✸ t❛ ❝ã SuppR HIi (M ) ❱× t❤Õ✱ t❤❡♦ ❣✐➯ t❤✐Õt t❛ ❝ã Pr = i=0 dim(R/p) s ✈í✐ ♠ä✐ p ∈ Pr Số hóa trung tâm học lieäu ❈❤♦ p ∈ T r (I, M ) ≥s ∪ T r (a, M ) http://www.lrc-tnu.edu.vn/ ≥s ✸✼ ◆Õ✉ dim(R/p) > s t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✷✳ p p ∈ / Pr ◆Õ✉ p ∈ AssR ExtrR (R/I, M ) ❙✉② r❛ dim(R/p) = s ❇ỉ ➤Ị ✷✳✹✳✷✳ ❈❤♦ r❛ t❤× ❚❛ ❝ã p ∈ AssR ExtrR (R/I, M ) ∪ Pt p ∈ / AssR ExtrR (R/I, M ) ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❝đ❛ ♥➭♦ ➤ã✳ ❱× t❤Õ p ∈ AssR (ExtiR (R/I, M )) r ♥➭♦ ➤ã✳ ❱× ✈❐②✱ T (I, M ) ✈➭ ≥s r AssR ExtiR (R/I, M ) i=0 t❤❡♦ t❤× p ∈ Pr ✱ ExtiR (R/I, M ) ✈➭ s✉② ✈í✐ i < r ✈í✐ ♠ét sè ♥❣✉②➟♥ i < r T r (a, M ) ≥s ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛ t❐♣ ➜Þ♥❤ ❧Ý s❛✉ ➤➞② ❧➭ ❦Õt q✉➯ ❝❤Ý♥❤ t❤ø ❤❛✐ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥➭②✳ s, t ∈ N ✷✳✹✳✻ ➜Þ♥❤ ❧ý✳ ❈❤♦ ✈í✐ ♠ä✐ ✈í✐ t ≥ 1✳ ●✐➯ sö dim(Supp(HIi (M ))) i < t ●ä✐ (x1 , , xt ) ❧➭ ❞➲② ❝➳❝ ❝➳❝ ♣❤➬♥ tö tr♦♥❣ I s s❛♦ ❝❤♦ ♥ã M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ > s ✈õ❛ ❧➭ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ✈õ❛ ❧➭ ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭♥❤÷♥❣ ❞➲② ♥❤➢ t❤Õ ❧➭ tå♥ t➵✐ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✸✳✶✵✮✳ ❑❤✐ a = (a1 , , ak ) ❝đ❛ I ➤ã ✈í✐ ♠ä✐ ❤Ư s✐♥❤ ❝➳❝ t❐♣ T r (I, M ) ≥s ✈➭ T r (a, M ) ≥s ✈➭ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ r t✱ ❝❤ø❛ tr♦♥❣ t❐♣ ❤÷✉ ❤➵♥ r AssR M/(x1 , , xr )M ≥s+1 ∪ AssR M/(x1 , , xi )M s i=0 ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ i < r ❱× r t✳ ❈❤♦ p ∈ T r (I, M ) dim(SuppR HIi (M )) SuppR HIi (M ) ≥s s ≥s SuppR HIi (M ) SuppR HIi (M )✱ HIi (M ) ❙✉② r❛ ✈í✐ ♠ä✐ i < r ≥s = AssR HIi (M ) s ❱× t❤Õ✱ t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✷ ✈➭ ❇ỉ ➤Ị ✷✳✹✳✸ t❛ s✉② r❛ r−1 AssR HIi (M ) p∈ s ∪ AssR HIr (M ) ≥s i=0 Số hóa trung tâm học lieọu s sử ỗ tè tr♦♥❣ ➤Ị✉ ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ ♥ã ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ∪ T r (a, M ) http://www.lrc-tnu.edu.vn/ ✈➭ ✈× t❤Õ t❛ ❝ã ✸✽ (x1 , , xt ) ∈ I I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ ♥➟♥ t❤❡♦ i ❇ỉ ➤Ị ✷✳✸✳✽ t❛ ❝ã H (M ) ∼ (M )) ✈í✐ ♠ä✐ i = 1, , r ❱× = H (H i ❱× ❧➭ ột I tế ỗ I i = 1, , r (x1 , ,xi )R t❛ ❝ã i AssR HIi (M ) ⊆ AssR H(x (M ) ⊆ , ,xi )R AssR M/(xn1 , , xni )M n∈N ❱× (x1 , , xt ) ❧➭ M ✲❞➲② ❦❤➠♥❣ ➤✐Ò✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ > s ♥➟♥ AssR M/(xn1 , , xni )M s = AssR (M/(x1 , , xi )M ) s n∈N ♠ä✐ i = 1, , r ✭t❤❡♦ ▼Ö♥❤ ➤Ò ✷✳✸✳✶✶✮ ✈➭ AssR M/(xn1 , , xnr )M ≥s+1 = AssR M/(x1 , , xr )M ≥s+1 n∈N ◆Õ✉ dim(R/p) > s r = t t❤× p ∈ AssR (M/(x1 , , xr )M ) ✈➭ ≥s+1 r ❱× t❤Õ t❛ s✉② r❛ ◆Õ✉ dim(R/p) = s t❤× r AssR (HIi (M )) s p∈ p ∈ AssR HIr (M )✳ = i=0 AssR M/(x1 , , xi )M s i=0 ❱❐② ➜Þ♥❤ ❧Ý ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❍Ư q✉➯ s❛✉ ➤➞② s✉② r❛ ♥❣❛② tõ ➜Þ♥❤ ❧Ý ✷✳✹✳✺ ✈➭ ➜Þ♥❤ ❧Ý ✷✳✹✳✻✳ ✷✳✹✳✼ ❍Ư q✉➯✳ ❈❤♦ 1✱ t ∈ N s ✈➭ dim(SuppR HIi (M )) s ✈í✐ ♠ä✐ i < t ❈❤♦ a = (a1 , , ak ) ❧➭ ột ệ s ủ I ó ỗ sè tù ♥❤✐➟♥ r t✱ ❝➳❝ t❐♣ ❤ỵ♣ T r (I, M ) ✈➭ T r (a, M ) ❝❤ø❛ r AssR ExtiR (R/I, M ) ∪ {m} tr♦♥❣ t❐♣ ❤÷✉ ❤➵♥ i=0 ✭✐✐✮ ❈❤♦ (x1 , , xt ) I I ✲❞➲② tr♦♥❣ ♥❣✉②➟♥ ✈➭ ❧➭ r ❧➭ M ✲❞➲② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ❧ä❝ ❝❤Ý♥❤ q✉② ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ sè t, ❝➳❝ t❐♣ T r (I, M ) ✈➭ T r (a, M ) ❝❤ø❛ tr♦♥❣ t❐♣ ❤÷✉ ❤➵♥ r AssR M/(x1 , , xr )M ≥s+1 ∪ AssR M/(x1 , , xi )M i=0 Số hóa trung tâm học lieäu http://www.lrc-tnu.edu.vn/ s ∪ {m} ✸✾ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ♥➭② tr×♥❤ ❜➭② ❧➵✐ ♠ét sè ❦Õt q✉➯ ➤➲ ❜✐Õt ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ Ext✳ ▲✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ s❛✉✿ ✶✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✈Ị ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ♠➠➤✉♥ ◆♦❡t❤❡r✱ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ♠➠➤✉♥ ♠ë ré♥❣ ✈➭ ❝❤✐Ị✉ ❑r✉❧❧✳ ✷✳ ◆❤➽❝ ❧➵✐ ❝➳❝ ❦Õt q✉➯ ✈Ò ❞➲② ❝❤Ý♥❤ q✉② ✈➭ tr×♥❤ ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t ✈Ị ❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ✸✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ♠è✐ q✉❛♥ ❤Ư ✈í✐ ❞➲② ❝❤Ý♥❤ q✉② t❤❡♦ ❝❤✐Ị✉ ❧í♥ ❤➡♥ s ✹✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠➠➤✉♥ Số hóa trung tâm học liệu Ext http://www.lrc-tnu.edu.vn/ ✹✵ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬❇❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ♦❢ AssR (M/I n M ), Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✶✮ ✼✹ ✭✶✾✼✾✮✱ ✶✻✲✶✽✳ ❬❇◆❪ ▼✳ ❇r♦❞♠❛♥♥✱ ▲✳ ❚✳ ◆❤❛♥✱ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ Ext✲♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✻ ✭✷✵✵✽✮✱ ✶✺✷✼✲✶✺✸✻✳ ❬❇❙❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❘✳ ❨✳ ❙❤❛r♣✱ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦✲ ❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜✳ ❯♥✐✈✳ Pr❡ss✱ ✶✾✾✽✳ ❬❈❙❚❪ ◆✳ ❚✳ ❈✉♦♥❣✱ P✳ ❙❝❤❡♥③❡❧✱ ◆✳ ❱✳ ❚r✉♥❣✱ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ▼♦❞✉❧♥✱ ▼❛t❤✳ ◆❛❝❤r✱ ✽✺ ✭✶✾✼✽✮✱ ✺✼✲✼✸✳ ❬❑❛t❪ ▼✳ ❑❛t③♠❛♥✱ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐♥❢✐♥✐t❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❛ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✺✷ ✭✷✵✵✷✮✱ ✶✻✶✲✶✻✻✳ ❬❑❑❪ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤ ❛♥❞ ❋✳ ❑❤♦s❤✲❆❤❛♥❣✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ❝❡rt❛✐♥ s❡ts ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ ❊①t✲♠♦❞✉❧❡s✱ ▼❛♥✉s❝✐♣t❛ ▼❛t❤✳✱ ✶✷✺ ✭✷✵✵✽✮✱ ✸✹✺✲✸✺✷✳ ❬▼❛t❪ ❍✳ ▼❛ts✉♠✉r❛✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜✳ ❯♥✐✳ Pr❡ss✱ ✶✾✽✻✳ ❬▼❙❪ ▲✳ ▼❡❧❦❡rss♦♥ ❛♥❞ P✳ ❙❝❤❡♥③❡❧✱ ❆s②♠♣t♦t✐❝ ♣r✐♠❡ ✐❞❡❛❧s r❡❧❛t❡❞ t♦ ❞❡r✐✈❡❞ ❢✉♥❝t♦rs✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✹✮ ✶✶✼ ✭✶✾✾✸✮✱ ✾✸✺✲✾✸✽✳ ❬◆❙❪ ❯✳ ◆❛❣❡❧ ❛♥❞ P✳ ❙❝❤❡♥③❡❧✱ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ❛♥♥✐❤✐❧❛t♦rs ❛♥❞ ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ r❡❣✉❧❛r✐t②✱ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ✶✺✾✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✶✾✾✹✮✱ ✸✵✼✲✸✷✽✳ ❬◆❤❪ ▲✳ ❚✳ ◆❤❛♥✱ ❖♥ ❣❡♥❡r❛❧✐③❡❞ r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✸ ✭✷✵✵✺✮✱ ✼✾✸✲✽✵✻✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ... tÝ❝❤ tr➟♥✮ ✈➭ ❝➳❝ Số hóa trung tâm học liệu Qi pi Qi ❧➭ ❦❤➠♥❣ t❤õ❛ ✭tø❝ ❧➭ ➤➠✐ ♠ét ♣❤➞♥ ❜✐Ưt✳ http://www.lrc-tnu.edu.vn/ ✸ ❈❤ó ý r➺♥❣ ♥Õ✉ √ I ❧➭ ♥❣✉②➟♥ s➡ t❤× I ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tố ề ợ ú tứ ó... t❤❡♦ r ✈➭ sư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✹✳✶ ổ ề ổ ề ỗ số ♥❣✉②➟♥ r ≥ t❛ ❝ã r r SuppR ExtiR (R/I n , M ) SuppR ExtiR (R/I, M ) = i=0 n∈N i=0 r SuppR HIi (M ) = i=0 Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/... ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✱