❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ✣■◆❍ ❍Ú❯ ❉❯❨ ▼➷✣❯◆ P❍❹◆ ❙➮ ❙❯❨ ❘❐◆● ❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✷✷ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ✣■◆❍ ❍Ú❯ ❉❯❨ ▼➷✣❯◆ P❍❹◆ ❙➮ ❙❯❨ ❘❐◆● ❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ◆❣➔♥❤✿ ✣❸■ ❙➮ ❱⑨ ▲➑ ❚❍❯❨➌❚ ❙➮ số ữớ ữợ ✐ ▼ö❝ ❧ö❝ ▼ö❝ ❧ö❝ ❉❛♥❤ ♠ö❝ ❝→❝ ❦➼ ❤✐➺✉ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✣ë ❞➔✐ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤✐➲✉ ❑r✉❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤↕♠ trò ✈➔ ❤➔♠ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✶ P❤↕♠ trò ✈➔ ❤➔♠ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✷ ❍➔♠ tû a✲①♦➢♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ố ỗ ổ ố ỗ ữỡ ổ ố ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ố ỗ ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ●✐ỵ✐ ❤↕♥ t❤✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✷✳ ▼æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✐✐ ✶ ✸ ✸ ✹ ✻ ✶✵ ✶✵ ✶✷ ✶✹ ✶✹ ✶✼ ✷✷ ✷✻ ✷✳✶ ▼æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ▼ët sè t➼♥❤ ❝❤➜t ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ố ỗ ữỡ tt ỡ t❤ù❝ ✹✽ ✸✳✶ ▼æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✈➔ ✤è✐ ỗ ữỡ ✳ ✹✽ ✸✳✷ Ù♥❣ ❞ư♥❣ ✤è✐ ✈ỵ✐ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✵ ✻✷ ✐✐ ❉❛♥❤ ♠ö❝ ❝→❝ ❦➼ ❤✐➺✉ Z N0 Z+ R (R, m) ▼♦❞ (R) ❆ssR (M ) ❞✐♠R ❞✐♠M Γa (•) F (•) H n (•) Rn F Han (•) lim (•) −→ i U U −n M U −n (•) ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❱➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ❱➔♥❤ ◆♦❡t❤❡r ữỡ ợ ỹ m P trị ❝→❝ R−♠ỉ✤✉♥ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ R−♠æ✤✉♥ M ❈❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤ R ❈❤✐➲✉ ❑r✉❧❧ Rổ M tỷ a tữỡ ự ợ a ❍➔♠ tû tø ♣❤↕♠ trò ▼♦❞ (R) ✤➳♥ ❝❤➼♥❤ õ tỷ ố ỗ tự n tứ trò ▼♦❞ (R) ✤➳♥ ❝❤➼♥❤ ♥â ❍➔♠ tû ❞➝♥ ①✉➜t tự n tỷ F tỷ ố ỗ ữỡ tự n tữỡ ự ợ a ❍➔♠ tû ❣✐ỵ✐ ❤↕♥ t❤✉➟♥ ❚➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Rn ợ n số ữỡ ổ số s✉② rë♥❣ ❝õ❛ R−♠ỉ✤✉♥ M ù♥❣ ✈ỵ✐ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ U ❝õ❛ Rn ❍➔♠ tû tø ♣❤↕♠ trò ▼♦❞ (R) õ tữỡ ự ợ t t ❣✐→❝ U ❝õ❛ Rn ✶ ▼ð ✤➛✉ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ S ❧➔ ♠æt t➟♣ ♥❤➙♥ ✤â♥❣ ❝õ❛ ✈➔♥❤ R✱ M ❧➔ ♠ët R−♠ỉ✤✉♥✳ ❳➙② ❞ü♥❣ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ M × S ✿ ∀ (a, s) , (b, t) ∈ M × S, (a, s) ∼ (b, t) ⇔ u (ta − sb) = ✈ỵ✐ ♠ët u ∈ S tữỡ M ì S/ ữủ S −1 M = a s a ∈ M, s ∈ S ❑❤✐ ✤â S −1M ❧➔ ♠ët R−♠æ✤✉♥✱ ợ t ữủ a b , ∈ S −1 M, s t ✈➔ ∀ r ∈ R, ∀ a b ta + sb + = s t st a ∈ S −1 M, s a r = s s ▼æ✤✉♥ S −1M ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ♣❤➙♥ sè ✈➔ ♥â ❧➔ ♠ët tr♦♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè ❣✐❛♦ ❤♦→♥✳ ◆➠♠ ✶✾✽✷✱ ❘✳❨✳❙❤❛r♣ ✈➔ ❍✳❩❛❦❡r✐ ❬✶✽❪ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ U ❝õ❛ Rn = R × × R ✈ỵ✐ n ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ ①➙② ❞ü♥❣ ♠ỉ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ U −nM ✱ ♠é✐ ♣❤➛♥ tû ❝õ❛ ♥â ❝â ❞↕♥❣ m , (u1 , , un ) tr♦♥❣ ✤â m ∈ M ✈➔ (u1, , un) ∈ U ✳ ❑❤✐ n = 1✱ U ❧➔ ♠ët t➟♣ ♥❤➙♥ ✤â♥❣ ✈➔ U −1 M ổ số ỵ tt ổ số s✉② rë♥❣ ❧➔ ♠ët sü ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ♠æ✤✉♥ ♣❤➙♥ sè ❝õ❛ ♠ët ♠æ✤✉♥ t❤❡♦ t➟♣ ♥❤➙♥ ✤â♥❣ ✈➔ ♥â ✤÷đ❝ sû ❞ư♥❣ ✤➸ t✐➳♣ ❝➟♥ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ❝õ❛ ❍♦❝❤st❡r ❬✺❪✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ✤↕✐ sè ❣✐❛♦ ❤♦→♥ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐✿ ▼æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✈➔ ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❬✶✽❪✱ ❬✶✾❪✳ ▲✉➟♥ ✈➠♥ ♥❣♦➔✐ ♣❤➛♥ ▼ö❝ ❧ö❝✱ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❉❛♥❤ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝â ✸ ❝❤÷ì♥❣✳ ✷ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♥❤÷✿ ✣ë ❞➔✐ ♠ỉ✤✉♥✱ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✱ ❈❤✐➲✉ ❑r✉❧❧✱ P❤↕♠ trị ✈➔ ❤➔♠ tỷ ổ ố ỗ ổ ố ỗ ữỡ ợ t ữỡ ổ số s rở r ữỡ trữợ t ú tổ tr ❜➔② ❦❤→✐ ♥✐➺♠ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ✈➔ ①➙② ❞ü♥❣ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ❝õ❛ ♠ët ♠æ✤✉♥ t❤❡♦ ♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✈➔ ♠ët sè ✈➼ ❞ö ✈➲ t➟♣ ❝♦♥ t❛♠ ❣✐→❝✳ ữỡ ổ số s rở ố ỗ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣✱ ✈➔ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ♠é✐ ♠ỉ✤✉♥ M tr➯♥ tr ữỡ (R, m) ợ R = n ổ ố ỗ Hmn (M ) ❝â t❤➸ ①❡♠ ❧➔ ♠ët ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣✳ ❚✐➳♣ t❤❡♦ ✤â ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝❤♦ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ sỹ ữợ ú ù t t t ữợ ỏ rữớ ◗✉② ◆❤ì♥✳ ❚ỉ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ✤➣ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ qỵ rữớ ỡ Pỏ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯ ũ qỵ t ổ ợ ✣↕✐ sè ✈➔ ▲➼ t❤✉②➳t sè ❦❤â❛ ✷✸ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❚æ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ♥❣÷í✐ t❤➙♥✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❣✐ó♣ ✤ï ✤ë♥❣ ✈✐➯♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ò ữủ tỹ ợ sỹ ộ ỹ ố ❣➢♥❣ ❤➳t sù❝ ❝õ❛ ❜↔♥ t❤➙♥✱ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚ỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ỳ õ ỵ qỵ t ổ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✸ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✤➸ ❝❤✉➞♥ ❜à ❝❤♦ ♥ë✐ ❞✉♥❣ ❝→❝ ❝❤÷ì♥❣ t✐➳♣ t❤❡♦✳ ✶✳✶ ✣ë ❞➔✐ ♠ỉ✤✉♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✤ë ❞➔✐ ♠æ✤✉♥ ✈➔ ♠ët sè ❦➳t q✉↔ ✈➲ ✤ë ❞➔✐ ♠æ✤✉♥ t❤❡♦ ❬✶❪✱ ❬✸❪✱ ❬✶✸❪✳ ❑➼ ❤✐➺✉ R ❧➔ ♠ët ✈➔♥❤ ❝â ✤ì♥ ✈à✱ Z+ ❧➔ t➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët R−♠ỉ✤✉♥ M ❦❤→❝ ❦❤ỉ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ✤ì♥ ♥➳✉ ♥â ❝â ✤ó♥❣ ❤❛✐ ♠ỉ✤✉♥ ❝♦♥ ❧➔ ♠æ✤✉♥ ❦❤æ♥❣ ✈➔ ❝❤➼♥❤ ♥â✳ ❇ê ✤➲ ✶✳✶✳✷✳ ❈❤♦ M ❧➔ ♠ët R−♠ỉ✤✉♥✳ ❑❤✐ ✤â M ❧➔ R−♠ỉ✤✉♥ ✤ì♥ M = R/m ữ Rổ ợ m ∈ Max (R)✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ▼ët ❞➙② ❝❤✉②➲♥ ❝❤➦t ❝â ✤ë ❞➔✐ n ❝õ❛ R−♠æ✤✉♥ M ❧➔ ♠ët ❞➣② t➠♥❣ t❤ü❝ sü ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❝â ❞↕♥❣ M0 ⊊ M1 ⊊ ⊊ Mn✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ▼ët ❞➙② ❝❤✉②➲♥ ❝❤➦t ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M ❝â ❞↕♥❣ = M0 ⊊ M1 ⊊ ⊊ Mn = M ✱ tr♦♥❣ ✤â Mi /Mi−1 ❧➔ ♠ỉ✤✉♥ ✤ì♥ ∀i = 1, , n ✭tù❝ ❧➔ ❞➣② ❦❤æ♥❣ t❤➸ ❜ê s✉♥❣ t❤➯♠✮✱ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉é✐ ❤đ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ n ❝õ❛ ♠ỉ✤✉♥ M ✳ ▼ỉ✤✉♥ ❦❤ỉ♥❣ ✤÷đ❝ ❝♦✐ ❧➔ ❝â ❝❤✉é✐ ❤ñ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ ❜➡♥❣ ỵ ỵ rr M ♠ët R−♠ỉ✤✉♥✳ ●✐↔ sû r➡♥❣ M ❝â ♠ët ❝❤✉é✐ ❤đ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ n✳ ❑❤✐ ✤â✱ ✭✐✮ ▼å✐ ❞➙② ❝❤✉②➲♥ ❝❤➦t ❝õ❛ M ✤➲✉ ❝â ✤ë ❞➔✐ ❦❤ỉ♥❣ ❧ỵ♥ ❤ì♥ n✳ ✭✐✐✮ ▼å✐ ❝❤✉é✐ ❤đ♣ t❤➔♥❤ ❝õ❛ M ✤➲✉ ❝â ✤ë ❞➔✐ ✤ó♥❣ ❜➡♥❣ n✳ ✭✐✐✐✮ ▼å✐ ❞➙② ❝❤✉②➲♥ ❝❤➦t ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❝â ✤ë ❞➔✐ k < n ✤➲✉ ❝â t❤➸ ❜ê s✉♥❣ n − k t❤➔♥❤ ♣❤➛♥ ✤➸ trð t❤➔♥❤ ♠ët ❝❤✉é✐ ❤ñ♣ t❤➔♥❤ ❝õ❛ M ✳ ✭✐✈✮ ▼å✐ ❞➙② ❝❤✉②➲♥ ❝❤➦t ❝õ❛ M ❝â ✤ë ❞➔✐ ✤ó♥❣ ❜➡♥❣ n ✤➲✉ ❧➔ ❝❤✉é✐ ❤đ♣ t❤➔♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ❑❤✐ R−♠ỉ✤✉♥ M ❝â ♠ët ❝❤✉é✐ ❤đ♣ t❤➔♥❤ ❝â ✤ë ❞➔✐ n < ∞ t❤➻ t❛ ♥â✐ M ❝â ✤ë ❞➔✐ ❜➡♥❣ n ✈➔ ❦➼ ❤✐➺✉ lR (M ) = n✳ ✹ ❱➼ ❞ö ✶✳✶✳✼✳ ✶✳ ❈❤♦ V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tr➯♥ tr÷í♥❣ K ❑❤✐ ✤â✱ V ❝â ❝❤✐➲✉ ❤ú✉ ❤↕♥ ⇔ V ❧➔ K−♠æ✤✉♥ ◆♦❡t❤❡r ⇔ V ❧➔ K−♠ỉ✤✉♥ ❆rt✐♥✳ ❍ì♥ ♥ú❛✱ V ❧➔ R−♠ỉ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ lK (V ) = ❞✐♠K (V )✳ ✷✳ lZ (Z30) = 3✳ ●❤✐ ❝❤ó ✶✳✶✳✽✳ ✶✳ ▼ët R−♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ◆♦❡t❤❡r ♥➳✉ ♠å✐ ❞➣② t➠♥❣ M0 ⊆ M1 ⊆ Mn+1 ⊆ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ứ tự tỗ t k Z+ : Mk = Mk+i ✈ỵ✐ ♠å✐ i ∈ Z+ ✳ ▼ët ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ♥➳✉ R ❧➔ R−♠ỉ✤✉♥ ◆♦❡t❤❡r✳ ✷✳ ▼ët R−♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ❆rt✐♥ ♥➳✉ ♠å✐ ❞➣② ❣✐↔♠ M0 ⊇ M1 ⊇ Mn+1 ⊇ ❝→❝ ổ M ứ tự tỗ t k ∈ Z+ : Mk = Mk+i ✈ỵ✐ ♠å✐ i ∈ Z+ ✳ ▼ët ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët rt R Rổ rt ỵ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥✳ ❑❤✐ ✤â lR (M ) < ∞ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M ✈ø❛ ❧➔ ♠æ✤✉♥ tr ứ ổ rt ỵ −→ N −→ M −→ P −→ ❧➔ ♠ët ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ R−♠ỉ✤✉♥✳ ❑❤✐ ✤â✱ ✭✐✮ lR (M ) < ∞ ⇔ lR (N ) < ∞ ✈➔ lR (P ) < ∞✳ ✭✐✐✮ ❑❤✐ lR (M ) , lR (N ) , lR (P ) ✤➲✉ ❤ú✉ ❤↕♥ t❤➻ lR (M ) = lR (N ) + lR (P )✳ ✶✳✷ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤❡♦ ❬✸❪✱ ❬✶✸❪✱ ❬✶✹❪✳ ❑➼ ❤✐➺✉ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ▼ët ✐✤➯❛♥ I ❝õ❛ ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ ♥➳✉ I ⊊ R ✈➔ ✈ỵ✐ ♠å✐ a, b ∈ R, ab ∈ I t❤➻ a ∈ I ❤♦➦❝ bk ∈ I ✈ỵ✐ ♠ët k ∈ Z+✳ ●❤✐ ❝❤ó ✶✳✷✳✷✳ ✶✳ ▼é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè P ❝õ❛ R ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ sì✳ ✷✳ I ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè I⊊R ⇔ ∀ a, b ∈ R, ab ∈ I ⇒ a ∈ I √ ❤♦➦❝ b ∈ I √ ✸✳ ❈❤♦ I ✁ R✳ ❑❤✐ ✤â ♥➳✉ I √❧➔ ✐✤➯❛♥√❝ü❝ ✤↕✐ t❤➻ √I ❧➔ ♥❣✉②➯♥ sì✳ ✹✳ ●✐↔ sû m ∈ Max (R)✱ ✈➻ mk = m ∩ ∩ m = m ♥➯♥ mk ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ sì✳ ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ▼ỉ✤✉♥ ❝♦♥ t❤ü❝ sü N ❝õ❛ R−♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ ♥❣✉②➯♥ ♥➳✉ ∀α ∈ R, ∀x ∈ M, αx ∈ N ⇒ x ∈ N ❤♦➦❝ ∃k ∈ Z+ s❛♦ ❝❤♦ αk M ⊆ N ✳ ●❤✐ ❝❤ó ✶✳✷✳✹✳ ✶✳ ■✤➯❛♥ I ❝õ❛ R ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ I ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ R−♠ỉ✤✉♥ R✳ ✷✳ N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ❦❤✐ ✈➔ ợ R tỹ ỗ ❝➜✉ M/N −→ M/N ❤♦➦❝ ❧➔ ✤ì♥ ❝➜✉ ❤♦➦❝ ❧➔ ❧ô② ❧✐♥❤✳ ❇ê ✤➲ ✶✳✷✳✺✳ ✭✐✮ ❈❤♦ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ R−♠æ✤✉♥ M ✳ ❑❤✐ ✤â ❘❛❞M (N ) = α ∈ R | ∃k ∈ Z+ s❛♦ ❝❤♦ αk M ⊆ N ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ✣➦❝ ❜✐➺t✱ ♥➳✉ a ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R t❤➻ ❘❛❞R (a) = √a✳ ✭✐✐✮ ❈❤♦ N, P ❧➔ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ◆➳✉ N ⊆ P t❤➻ ❘❛❞M (N ) ⊆ ❘❛❞M (P )✳ ❍ì♥ ♥ú❛ ❘❛❞M (N ∩ P ) = ❘❛❞M (N ) ∩ ❘❛❞M (P ) ▼➺♥❤ ✤➲ ✶✳✷✳✻✳ ❈❤♦ N ❧➔ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ R−♠ỉ✤✉♥ M ✳ ❑❤✐ ✤â ❘❛❞M (N ) = P ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❚❛ ❣å✐ N ❧➔ ♠ỉ✤✉♥ ❝♦♥ P −♥❣✉②➯♥ ❝õ❛ M ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✼✳ ■✤➯❛♥ ♥❣✉②➯♥ tè P ❝õ❛ ✈➔♥❤ R ữủ t ợ Rổ M tỗ t x M s (x) = P ✳ ❚➟♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ❆ssR (M ) ❤❛② ✤ì♥ ❣✐↔♥ ❤ì♥ ❆ss (M ) ♥➳✉ R ✤÷đ❝ ①→❝ ✤à♥❤✳ P ∈ ❙♣❡❝ (R) ●❤✐ ❝❤ó ✶✳✷✳✽✳ ✶✳ P ∈ ❆ssR (M ) ⇔ P = ❆♥♥ (x) = :M x = m ∈ R | mx = ✷✳ ❈❤♦ P ∈ ❙♣❡❝ (R)✳ ❑❤✐ ✤â ♥➳✉ P ∈ ❆ss (M ) t❤➻ tỗ t ởt ổ N M s N ∼= R/P ✳ ✸✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R−♠æ✤✉♥✳ ✭✐✮ ●✐↔ sû M ̸= 0✳ ❑➼ ❤✐➺✉ F = ❆♥♥ (x) | x ∈ M \ {0} ✳ ❑❤✐ ✤â ♠å✐ ♣❤➛♥ tû tè✐ ✤↕✐ ❝õ❛ ❤å F ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè tù❝ ❧➔ P ∈ ❆ss (M )✳ ✭✐✐✮ ❆ss (M ) = ∅ ⇔ M = 0✳ ✭✐✐✐✮ ❩❉R (M ) = P✳ P ∈❆ss(M ) ▼➺♥❤ ✤➲ ✶✳✷✳✾✳ ❈❤♦ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ R−♠æ✤✉♥ M ✳ ❑❤✐ ✤â ❆ss (N ) ⊆ ❆ss (M ) ⊆ ❆ss (N ) ∪ ❆ss M/N ✻ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✵✳ ❈❤♦ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ R−♠æ✤✉♥ M ✳ ▼ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ N ữợ ổ sỡ ❝õ❛ M N = P ∩ P2 ∩ ∩ Pr , tr♦♥❣ ✤â P1, , Pr ❧➔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❝õ❛ M ✳ ●❤✐ ❝❤ó ✶✳✷✳✶✶✳ ✶✳ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ N = P1 ∩ P2 ∩ ∩ Pr ✤÷đ❝ ❣å✐ ❧➔ rót ❣å♥ ♥➳✉ Pk ⊈ Pi, ∀i = 1, r ✈➔ ❘❛❞M (Pi) ̸= ❘❛❞M Pj ✈ỵ✐ ⩽ i ̸= j ⩽ r✳ k̸=i ✷✳ ▼å✐ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ✤➲✉ õ t ữ rút ỵ ▼å✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ ♠ët ♠æ✤✉♥ ◆♦❡t❤❡r ✤➲✉ ❝â sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥✳ ▼➺♥❤ ✤➲ ✶✳✷✳✶✸✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ●✐↔ sû = N1 ∩ N2 ∩ ∩ Nr ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ rót ❣å♥ ❝õ❛ ♠ỉ✤✉♥ ❝♦♥ 0✳ ✣➦t Pi = ❘❛❞M (Ni) , ∀i = 1, r✳ ❑❤✐ ✤â ❆ss (M ) = {P1, , Pr }✳ ✶✳✸ ❈❤✐➲✉ ❑r✉❧❧ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ❑r✉❧❧ ✈➔ ♠ët sè ❦➳t q✉↔ t rữợ t ú tæ✐ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈➔ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝✳ ❑➼ ❤✐➺✉ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ tỗ t ởt ♥❤â♠ ❝♦♥ ❝ë♥❣ ❣✐❛♦ ❤♦→♥ (Rn)n⩾0 ❝õ❛ R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ R = Rn✳ n⩾0 ✭✐✐✮ ✈ỵ✐ ♠å✐ m, n ⩾ 0✳ ❱➼ ❞ư ✶✳✸✳✷✳ ✶✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ❜➜t ❦ý✳ ❈❤♦ R0 = R ✈➔ Rn = ✈ỵ✐ ♠å✐ n ⩾ 1✳ ❑❤✐ ✤â R ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈➔ ❣å✐ ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ t➛♠ t❤÷í♥❣✳ ✷✳ ❳➨t ✈➔♥❤ ✤❛ t❤ù❝ n ❜✐➳♥ R = K [x1, , xn] ✈ỵ✐ K ❧➔ ♠ët tr÷í♥❣✳ ●å✐ Rd ❧➔ t➟♣ t➜t ❝↔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❜➟❝ d✱ t➼♥❤ ❝↔ ✤❛ t❤ù❝ ❦❤æ♥❣✳ ❑❤✐ ✤â t❛ ❝â R= Rd ✈➔ Rd Rm ⊆ Rd+m ✈ỵ✐ ♠å✐ m, d ⩾ 0✳ ❱➟② R ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝✳ Rn Rm ⊆ Rm+n d⩾0 ✹✾ ❇ê ✤➲ ✸✳✶✳✶✳ ([19, 2.1])✳ ❈❤♦ Uu❧➔m♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Rn✱ ✈➔ ❣✐↔msû m ∈ M ✈➔ (u1, , un) ∈ U s❛♦ ❝❤♦ (u , n , u tr♦♥❣ U −nM ✳ ❑❤✐ ✤â (u , , u =0 n) n) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (u ,u.n m , u ) = 0✳ ❑❤✐ ✤â ∃ (w1, , wn) ∈ U ✈➔ H = [hij ] ∈ Dn (R) n s❛♦ ❝❤♦ Hu = w = Inw tr♦♥❣ ✤â u = [u1 un]T , w = [w1 wn]T ✈ỵ✐ = (w , , w ) n ✈➔ n−1 |H|un m = |H|un m − |In |0 ∈ Rwi M i=1 ❱➻ h11 · · · ✳✳✳ ✳ ✳ ✳ ✳✳✳ hn1 · · · hnn ♥➯♥ u1 ✳✳✳ un = w1 ✳✳✳ wn n−1 wn = hni ui + hnn un i=1 ❙✉② r❛ n−1 n−1 hii wn − i=1 n−1 hii hnn um ∈ hni ui m = i=1 n−1 Rwi M i=1 i=1 ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✸✱ t❛ ❝â n−1 n−1 hii i=1 n−1 hni ui m = i=1 n−1 ❙✉② r❛ h11 hn−1n wnm ∈ n−1 ❚❤❡♦ ●❤✐ ❝❤ó ✷✳✷✳✹✭✸✮✱ t❛ ❝â i=1 Rwi M ✳ i=1 h11 hn−1n−1 wn m = w1 , wn−1 , wn2 ❍ì♥ ♥ú❛ n−1 hii ui m ∈ i=1 Rwi M i=1 ✺✵ In w1 ✳✳✳ = wn2 w1 ✳✳✳ wn2 = ··· ✳✳✳ ✳ ✳ ✳ ✳✳✳ · · · wn w1 ✳✳✳ wn , T = ··· ✳✳✳ ✳ ✳ ✳ ✳✳✳ · · · wn ✈➔ n−1 hii wn m − |T| |In | n−1 n−1 ∈ hii wn m i=1 i=1 n−1 hii wn m − hii m = i=1 i=1 n−1 Rwi M i=1 ❙✉② r❛ n−1 hii wn m, w1 , , wn2 ∼ i=1 ❱➟② n−1 hi m, (w1 , , wn ) i=1 n−1 hii wn m i=1 ❉♦ ✤â ❍ì♥ ♥ú❛ w1 , , wn2 = h11 hnn m (w1 , , wn ) h11 hnn m = (w1 , , wn ) In w1 ✳✳✳ wn = ✈➔ w1 ✳✳✳ wn = H n−1 |In | |H|m − |H|m = ∈ u1 ✳✳✳ un Rwi M i=1 ❙✉② r❛ h11 hnn m, (w1 , , wn ) ∼ m, (u1 , , un ) ❱➟② h11 hnn m m = = (w1 , , wn ) (u1 , , un ) P❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t t❤ó❝✳ ✺✶ ❚r♦♥❣ ❜ê ✤➲ t✐➳♣ t❤❡♦✱ ❣✐↔ sû R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ ❝❤ó♥❣ t ũ Hai (ã) ợ i tỷ ố ỗ ữỡ tự i tữỡ ự ✈ỵ✐ ✐✤➯❛♥ a ❝õ❛ ✈➔♥❤ R t❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✷✳✾✳ ❇ê ✤➲ ✸✳✶✳✷✳ ([19, 2.2])✳ ●✐↔ sû R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ ❝❤♦ a ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✱ ❝❤♦ U ❧➔ ♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Rn s❛♦ ❝❤♦ un ∈ a ✈ỵ✐ ♠å✐ (u1, , un) ∈ U ✳ ❑❤✐ ✤â Haj U −nM = ✈ỵ✐ ♠å✐ j ⩾ 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ f = (f1, , fn) ∈ U ✳ ✣➦t Uf = f1α , , fnα αi ∈ Z+, ∀i = 1, n ✱ ❦❤✐ ✤â t❤❡♦ ❱➼ ❞ö ✷✳✷✳✺✱ Uf ❧➔ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Rn✱ R−♠ỉ✤✉♥ Uf−nM ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ Mf ✳ ❳➨t →♥❤ ①↕ Mf −→ Mf : Φfn n m α1 f1 , , fnαn −→ fn m α1 f1 , , fnαn m = Φfn α1 f1 , , fnn õ f ởt ỗ t n ❦❡r Φf = n m f1α1 , , fnαn fn m α1 f1 , , fnαn ∈ Mf ❚❤❡♦ ❇ê ✤➲ ✸✳✶✳✶✱ t❛ ❝â ❦❡r Φf = {0}✳ ❱ỵ✐ ♠å✐ f α , a , f α ∈ Mf ✱ t❛ ❝â f α , a , f α n n 1 ❚❤➟t ✈➟② =0 n n ··· ✳✳✳ ✳ ✳ ✳ ✳✳✳ · · · fn f1α1 ✳✳✳ fnαn n = f1α1 ✳✳✳ fnαn+1 = ✈➔ |In| (fna) − |T|a = ∈ n−1 Rfiαi M ✈ỵ✐ i=1 ❉♦ ✤â Φf a f1α1 , , fnαn+1 n = T = fn a α1 f1 , , fnαn+1 = In fn a α1 f1 , , fnαn+1 = n n n n ✳✳✳ fnαn+1 ✳✳✳ ✳ ✳ ✳ ✳✳✳ ✳ fn a f1α1 , , fnαn f Mf ❧➔ ✤ì♥ ❝➜✉ ❱➟② Φf ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ❱➻ Φf : Mf −→ f ♥➯♥ Γa Φf : Γa Mf −→ Γa Mf ❧➔ ✤ì♥ ❝➜✉✳ n f1α1 ✳ ✳ ✺✷ ❍ì♥ ♥ú❛✱ ✈➻ fn ∈ a ✈➔ Γa Mf ❧➔ a−①♦➢♥✱ Γa Γa Mf = 0✳ ❙✉② r❛✱ t❤❡♦ ●❤✐ ❝❤ó ✶✳✺✳✷✳✶✹✭✸✮ fn Mf −→ Γa Mf ❧➔ ✤ì♥ ❝➜✉ ♥➯♥ Ha0 Mf = Γa Mf = ❱➻ Φf ❧➔ ✤ì♥ ❝➜✉ ♥➯♥ fn ∈ ◆❩❉R n Mf ✳ ❚❛ ❝â ❞➣② ❦❤ỵ♣ ♥❣➢♥ fn → → Mf −→ Mf → ❑❤✐ ✤â✱ t❛ ❝â ❞➣② ❦❤ỵ♣ → Ha0 (0) → Ha0 Mf → Ha0 Mf fn → Ha1 (0) → Ha1 Mf −→ Ha1 Mf fn → Ha2 (0) → Ha2 Mf −→ Ha2 Mf fn → Haj (0) → Haj Mf −→ Haj Mf → ❙✉② r❛ ♣❤➨♣ ♥❤➙♥ fn ❧➔ ✤ì♥ ❝➜✉✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ●❤✐ ❝❤ó ✶✳✺✳✷✳✶✹✭✷✮ ✈➻ fn ∈ a ✈➔ Haj Mf ❧➔ a−①♦➢♥ ♥➯♥ Haj Mf = 0, ∀j ⩾ 1✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✼✱ t❛ ❝â U −n M ∼ M = lim −→ f f ∈U ✈➔ s✉② r❛ tø [17, 3.2] r➡♥❣ Haj U −nM = 0, ∀j ⩾ 0✳ P❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t t❤ó❝✳ ❚r♦♥❣ s✉èt ♠ư❝ t✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❣✐↔ sû R ❧➔ ởt tr ữỡ ợ ỹ ✤↕✐ m, ❞✐♠R = n (n ⩾ 1) ✈➔ M ❧➔ ♠ët R−♠ỉ✤✉♥✳ ❈❤ó♥❣ tỉ✐ ❞ị♥❣ ✧s✳♦✳♣✧ ✤➸ t❤❛② ❝❤♦ ✧❤➺ t❤❛♠ sè✧✱ ✧s✳s✳♦✳♣✧ ✤➸ t❤❛② ❝❤♦ ✧♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè✧✳ ●❤✐ ❝❤ó ✸✳✶✳✸✳ ✶✳ ❱ỵ✐ ♠é✐ i ∈ Z+✱ ✤➦t Ui = ✈ỵ✐ ⩽ j ⩽ i : ∃j ∈ N0 ✈➔ sj+1 = = si = (s1 , , si ) ∈ Ri s1 , , s j ❧➔ s✳s✳♦✳♣ ❚❤❡♦ ❱➼ ❞ư ✷✳✷✳✾ ✈➔ ●❤✐ ❝❤ó ✷✳✷✳✹✭✶✮✱ ❞➵ t❤➜② r➡♥❣ Ui ❧➔ ♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Ri ✈ỵ✐ ♠é✐ i ⩾ 1✳ ❈❤♦ ♠ët s✳♦✳♣ x1 , , xn ❝õ❛ R ❞♦ ✤â x = (x1 , , xn ) ∈ Un ✱ ❝❤ó♥❣ t❛ ✤➦t U (x)i ❧➔ ♠ð rë♥❣ ❬①❡♠ ✷✳✷✳✹✭✶✮❪ ❝õ❛ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ αj ∈ Z+ ✈ỵ✐ ♠å✐ j = 1, i ❝õ❛ Ri ✱ tr♦♥❣ ✤â xr = ❦❤✐ r > n✳ xα1 , , xi ú ỵ r U (x)i ⊂ Ui ✈ỵ✐ ♠å✐ i ∈ Z+✳ i ✺✸ ✷✳ ❈❤♦ (Vi)i∈Z ❧➔ ♠ët ❤å ♥❤ú♥❣ t➟♣ t❤ä❛ ♠➣♥ ✭✐✮ Vi ⊆ Ui ✈➔ Vi ❧➔ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Ri ✈ỵ✐ ♠é✐ i ∈ Z+✳ ✭✐✐✮ ◆➳✉ (u1, , ui) ∈ Vi ✈ỵ✐ < i ∈ Z+ t❤➻ (u1, , ui−1) ∈ Vi−1✳ ✭✐✐✐✮ ◆➳✉ (u1, , ui) ∈ Vi ✈ỵ✐ i ∈ Z+ t❤➻ (u1, , ui, 1) Vi+1 ỗ t↕✐ ♠ët s✳♦✳♣ y1, , yn ❝õ❛ R s❛♦ ❝❤♦ (y1, , yn) ∈ Vn✳ ✭✈✮ (1) ∈ V1✳ ◆❤➟♥ ①➨t r➡♥❣ ❤å t➟♣ U (x)i i∈Z t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t ✭✐✮ ✤➳♥ ✭✈✮ tr♦♥❣ ✭✷✮✳ ❚ø ❝→❝ ❞ú ❧✐➺✉ tr➯♥✱ ❝❤ó♥❣ t❛ ①➙② ❞ü♥❣ ♠ët ✤è✐ ♣❤ù❝ ✤÷đ❝ ♠ỉ t↔ tr♦♥❣ ❜ê s ([19, 3.2]) ỗ t ỳ Rỗ e0m: M V11M i1 ei : Vi−i M −→ Vi+1 M ✈ỵ✐ ♠é✐ i > s❛♦ ❝❤♦ e0 (m) = ✈ỵ✐ ♠å✐ m ∈ M ✈➔ ✈ỵ✐ (1) ♠å✐ i > + + ei m (u1 , , ui ) = m , (u1 , , ui , 1) ✈ỵ✐ ♠å✐ m ∈ M ✈➔ (u1, , ui) ∈ Vi✳ ❍ì♥ ♥ú❛✱ e−1 e0 ei −i−1 −→ M −→ V1−1 M −→ V2−2 M −→ −→ Vi−i M −→ Vi+1 M −→ ❧➔ ♠ët ✤è✐ ♣❤ù❝ ❝õ❛ ♥❤ú♥❣ Rỗ Rổ (V , M ) V (Vi)i1 ú ỵ r t ●❤✐ ❝❤ó ✷✳✷✳✹✭✷✮✱ Vi−iM = ✈ỵ✐ ♠å✐ i > n + 1✳ ❇ê ✤➲ ✸✳✶✳✺✳ ([19, 3.3])✳ ❈❤♦ i ❧➔ ♠ët sè ♥❣✉②➯♥ s❛♦ ❝❤♦ ⩽ i ⩽ n✳ ❑❤✐ ✤â ❞✐♠ ❦❡r ei/✐♠ ei−1 < n − i ❈❤ù♥❣ ♠✐♥❤✳ ❚r÷í♥❣ ❤đ♣ i = 0✱ t❛ ❝â ❦❡r e0/✐♠ e−1 ∼= ❦❡r e0 = m m = 1 = m sm = ✈ỵ✐ s ∈ V1 ❙✉② r❛ ❞✐♠❦❡r e0 < n ●✐↔ sû i > 0✳ ❈❤ó♥❣ t❛ ❝➛♥ ❝❤ù♥❣ tä r➡♥❣✱ ♥➳✉ ❞✐♠ R/p < n − i✳ p ∈ ❆ss ❦❡r ei/✐♠ ei−1 t❤➻ ✺✹ ❑❤✐ ✤â p = ❆♥♥ (a) ✈ỵ✐ a ∈ ❦❡r ei/✐♠ ei−1 ✈➔ a = (u , m , u ) i =0:a = r ∈ R = = r ∈ R ∈ = ✐♠ ei−1 ✐♠ ei−1 : (u , m , u ) ❱➻ (u , m , u ) ∈ ❦❡r ei ♥➯♥ ei i i m (u1 , , ui ) = ✳ m =0 (u1 , , ui , 1) r tỗ t H = [hrs] ∈ Di+1 (R) ✈➔ (t1, , ti+1) ∈ Vi+1 s❛♦ ❝❤♦ H [u1 ui 1]T = [t1 ti ti+1 ]T ✈➔ i |H|m ∈ Rtj M j=1 ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✸✱ t❛ ❝â h11 hiiti+1m ∈ ❚❤❡♦ ●❤✐ ❝❤ó ✷✳✷✳✹✭✸✮✱ i Rtj M ✳ j=1 h11 hii ti+1 m ti+1 m = ∈ (u1 , , ui ) (t1 , , ti ) ✐♠ ei−1 ❱➻ ✈➟②✱ ✈ỵ✐ ♠å✐ j = 1, , i✱ t❛ ❝â tj m h11 hii tj m = ∈ (u1 , , ui ) (t1 , , ti ) ✐♠ ei−1 ❙✉② r❛ Rt1 + Rti+1 ⊆ p✳ ❉♦ ✤â t1, , ti, ti+1 ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ R✳ ❑❤✐ i = n t❛ ❝â ♠➙✉ t❤✉➝♥✳ ❱➻ ❆ss ❦❡r en/✐♠ en−1 = ∅ ✈➔ ❞✐♠ ❦❡r en/✐♠ en−1 = −1 ❦❤✐ < i < n✱ ❦➳t q✉↔ s✉② r❛ tø [15, ❝❤÷ì♥❣ ■❱, ✣à♥❤ ❧➼ ✷]✳ ❍➺ q✉↔ ✸✳✶✳✻✳ ([19, 3.4])✳ ✭✐✮ ❱ỵ✐ ♠å✐ i = 0, , n, j Hm ❦❡r ei/✐♠ ei−1 = ✈ỵ✐ ♠å✐ j ⩾ n − i ✭✐✐✮ ❱ỵ✐ ♠å✐ i = 1, , n, j Hm Vi−i M = ✈ỵ✐ ♠å✐ j ⩾ ✺✺ ✱ ✈ỵ✐ ♠å✐ i = 0, , n ♥➯♥ t❤❡♦ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❱➻ ❞✐♠ ❦❡r ei/✐♠ ei−1 ●❤✐ ❝❤ó ✶✳✺✳✷✳✶✷✭✐✮✱ t❛ ❝â i Hm ❦❡r ei/✐♠ ei−1 < n−i =0 ✈ỵ✐ ♠å✐ j ⩾ n − i ✭✐✐✮ ❈❤♦ i ❧➔ ♠ët sè tü ♥❤✐➯♥ ✈ỵ✐ ⩽ i ⩽ n✱ ✈➔ ✤➦t Wi = (u1 , , ui ) ∈ Vi | u1 , , ui ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝õ❛ R ❚ø ●❤✐ ❝❤ó ✸✳✶✳✸✭✐✮ ✤➳♥ ✭✈✮✱ Wi ❧➔ ♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Ri tỗ t ởt Rỗ i : Wii M −→ Vi−i M ①→❝ ✤à♥❤ ❜ð✐ ϕi m (w1 , , wi ) = m (w1 , , wi ) ✈ỵ✐ ♠å✐ m ∈ M ✈➔ (w1, , wi) ∈ Wi✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ tä r➡♥❣ ϕi ❧➔ t♦➔♥ ❝➜✉✳ ❈❤♦ α = (u , m , u ) ∈ ViiM t ú tỗ t (y1, , yi) i tỗ t↕✐ (v1, , vi) ∈ Vi ✈➔ H, K = [krs] ∈ Di (R) s❛♦ ❝❤♦ Wi H [u1 ui ]T = [v1 vi ]T = K [y1 yn ]T r ✈➻ vr = krsys ✈ỵ✐ ♠é✐ r = 1, , i✱ ♥➯♥ (v1, , vi) ∈ Wi✳ s=1 ❙✉② r❛ |H|m α= ∈ ✐♠ ϕi (v , , v ) ❉♦ ✤â i Wi−i M ∼ = Vi−i M ✈➔ tø ❇ê ✤➲ ✸✳✶✳✷ t❛ ❝â ❦➳t q✉↔ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ −n−1 n (M )✳ ❍ì♥ ỳ Rỗ tỹ ỵ ([19, 3.5]) ❝â Vn+1 M ∼ = Hm ♥❤✐➯♥ −n−1 −n−1 θn+1 : Vn+1 M −→ Un+1 M ①→❝ ✤à♥❤ ❜ð✐ θn+1 m (v1 , , vn+1 ) = m , (v1 , , vn+1 ) ✈ỵ✐ ♠å✐ m ∈ M ✈➔ (v1, , vn+1) ∈ Vn+1 ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ❉♦ ✤â −n−1 n Un+1 M∼ (M ) = Hm ✺✻ ❈❤ù♥❣ ♠✐♥❤✳ ❑➼ ❤✐➺✉ U ❧➔ ❤å (Ui)i⩾1 ✈➔ ✤➦t C (U , M ) ❧➔ d−1 d0 d1 di −i−1 −→ M −→ U1−1 M −→ U2−2 M → · · · → Ui−i M −→ Ui+1 M ÃÃà ợ ộ i tỗ t ởt Rỗ i : ViiM UiiM m (v1 , , vi ) θi = m (v1 , , vi ) ✈ỵ✐ ♠å✐ m ∈ M ✈➔ (v1, , vi) ∈ Vi✳ ◆➳✉ ✤➦t θ0 : M → M ỗ t t = i : C (V , M ) −→ C (U , M ) i0 ởt ỗ ố ♣❤ù❝✳ ✣➸ t✐➺♥ ❧ñ✐✱ ❦➼ ❤✐➺✉ C (V , M ) ❜ð✐ C (V ) ✈➔ C (U , M ) ❜ð✐ C (U )❀ V0−0M, U0−0M ❧➔ M ✳ ❱ỵ✐ ♠é✐ i = 0, , n✱ ✤➦t ❦❡r ei, ❦❡r di, K i = Vi−i M/ Li = Ui−i M/ ❝♦❦❡r ei−1 −→ ❝♦❦❡r di−1, θi : H i C (V ) −→ H i C (U ) ✈➔ θi+ : K i −→ Li ❧➔ ỳ ỗ t t tỗ t ỳ ỗ ợ ❦❤ỵ♣✮✳ ′ θi : ∗ ✴ ✴ V −i M K i−1 + ✴ ✎ ✴ i θi−1 ✴ Li−1 ✎ ❝♦❦❡r ei−1 θi ✴ Ui−i M ✎ θi ✴ ✴ ′ ❝♦❦❡r di−1 (1) ✈ỵ✐ ♠å✐ i = 1, , n ✈➔ ✴ 0 ✴ ✴ H i C (V ) ✎ θi ❝♦❦❡r ei−1 ∗ ✴ H i C (U ) ✎ θi ✴ ′ ✎ ❝♦❦❡r di−1 ✴ Ki θi ✴ Li + ✴ (2) ✈ỵ✐ ♠å✐ i = 0, , n − 1✳ ứ sỡ ỗ q ♠ët ❤➻♥❤ ✈✉æ♥❣ ❣✐❛♦ ❤♦→♥ n−i Hm n−i Hm ❝♦❦❡r ei−1 ✎ n−i Hm θi ∼ = ✴ n−i+1 K i−1 Hm ′ ❝♦❦❡r di−1 ∼ = ✴ ✎ n−i+1 Hm (θi−1+ ) n−i+1 Li−1 Hm (3) ✺✼ tr♦♥❣ ✤â ❤❛✐ ❤➔♥❣ ❧➔ ♥❤ú♥❣ ✤➥♥❣ ❝➜✉✱ ✈ỵ✐ ♠é✐ i = 1, , n✳ ❚÷ì♥❣ tü tø ỗ q ú t ữủ ❤➻♥❤ ✈✉æ♥❣ ❣✐❛♦ ❤♦→♥ = n−i ❝♦❦❡r ei−1 ∼ ✴ H n−i K i Hm m ′ ✎ n−i Hm θi ✎ n−i+1 Hm (θi−1+ ) (4) ❝♦❦❡r di−1 ∼= ✴ Hmn−i Li tr♦♥❣ ✤â ❤❛✐ ❤➔♥❣ ❧➔ ♥❤ú♥❣ ✤➥♥❣ ❝➜✉✱ ✈ỵ✐ ♠å✐ i = 0, , n − 1✳ ❚ø ❝→❝ ❤➻♥❤ ✈✉æ♥❣ ✭✹✮ ✈➔ ✭✸✮✱ ♥❤➟♥ ✤÷đ❝ ❤➻♥❤ ✈✉ỉ♥❣ ❣✐❛♦ ❤♦→♥ n−i Hm n Hm ❝♦❦❡r e−1 ∼ = ✴ ❝♦❦❡r en−1 Hm ′ n Hm ✎ n Hm θ0 ❝♦❦❡r ∼ = ✴ d−1 ✎ Hm θn ′ ❝♦❦❡r dn−1 Hm (5) tr♦♥❣ ✤â ❝→❝ ❤➔♥❣ ❧➔ ♥❤ú♥❣ ✤➥♥❣ ❝➜✉✳ ❚✉② ♥❤✐➯♥✱ tø ●❤✐ ❝❤ó ✸✳✶✳✸✱ ❇ê ✤➲ ✸✳✶✳✹✱ ✸✳✶✳✺ s✉② r ỗ n+1 Vn1 M en1 ✴ θn−1 −n+1 Un−1 M en ✴ Vn−n M dn−1 ✴ ✎ −n−1 e Vn+1 M θn dn ✴ Un−n M ✎ n+1 ✴ ✴ θn+1 −n−1 d Un+1 M n+1 (6) −n−1 ❝â ❝→❝ ❤➔♥❣ ❧➔ ❦❤ỵ♣✳ ❍ì♥ ♥ú❛✱ t❤❡♦ 18, 3.3✭✐✐✮ ♠é✐ ♣❤➛♥ tû ❝õ❛ Vn+1 M ❜à tr✐➺t −n−1 t✐➯✉ ❜ð✐ ♠ët ❧ô② t❤ø❛ ❝õ❛ m✱ ✈➔ ❦❤➥♥❣ ✤à♥❤ ✈➝♥ ✤ó♥❣ ✤è✐ ✈ỵ✐ Un+1 M ✳ ❚ø ✭✻✮ ❝❤♦ t❛ ♠ët ❤➻♥❤ ✈✉æ♥❣ ❣✐❛♦ ❤♦→♥ Hm ❝♦❦❡r en−1 ∼ = ✴ −n−1 M Vn+1 ′ Hm ✎ Hm θn ❝♦❦❡r dn−1 ∼ = ✴ ✎ θn+1 −n−1 Un+1 M (7) tr♦♥❣ ✤â ❤❛✐ ❤➔♥❣ ❧➔ ♥❤ú♥❣ ✤➥♥❣ ❝➜✉✳ ❱➻ θ0 = ✐❞M ✱ ♥➯♥ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ❤➻♥❤ ✈✉ỉ♥❣ ❣✐❛♦ ❤♦→♥ tø ✭✺✮ ✈➔ ✭✼✮ n (M ) Hm ✎ ∼ = ✴ n Hm (✐❞M ) (M ) Hm ∼ = ✴ −n−1 Vn+1 M ✎ θn+1 −n−1 Un+1 M (8) tr♦♥❣ ✤â ❝→❝ ❤➔♥❣ ❧➔ ♥❤ú♥❣ ✤➥♥❣ ❝➜✉✳ P❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t t❤ó❝✳ ✺✽ ❈❤♦ (R, m) ởt tr ữỡ ợ ❝ü❝ ✤↕✐ m ✈➔ ❞✐♠R = n ⩾ 1✳ ●✐↔ sû x = (x1, , xn) ❧➔ ♠ët ❤➺ t❤❛♠ sè ❜➜t ❦➻ ❝õ❛ R✳ ❈❤å♥ V = U (x)✱ ✈ỵ✐ ❝→❝ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ✸✳✶✳✸✱ U (x) = U (x)i i∈Z ✳ ❑❤✐ ✤â ♠æ✤✉♥ ✤è✐ ỗ ữỡ tự n Hmn (M ) õ t❤➸ ①❡♠ ♥❤÷ ❧➔ ♠ỉ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ t❤ỉ♥❣ q✉❛ ❤➺ t❤❛♠ sè x ❝õ❛ R✱ ✤✐➲✉ ✤â ✤÷đ❝ t❤➸ ❤✐➺♥ tr♦♥❣ ❦➳t q✉↔ s❛✉✳ ❍➺ q✉↔ ✸✳✶✳✽✳ ([19, 3.6])✳ ❈❤♦ x = (x1, , xn) ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ R✳ ❑❤✐ ✤â + −n−1 n (M ) ∼ Hm = U (x)n+1 M, tr♦♥❣ ✤â U (x)n+1 = xα1 , , xnn , tỗ t j ∈ N0 ✈ỵ✐ ⩽ j ⩽ n s❛♦ ❝❤♦ α1 , , αj ∈ Z+ ✈➔ αj+1 = = αn = n1 ỡ ỳ Rỗ tỹ ♥❤✐➯♥ U (x)−n−1 n+1 −→ Un+1 M ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ✸✳✷ Ù♥❣ ❞ư♥❣ ✤è✐ ✈ỵ✐ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ♠ö❝ ♥➔② ❝❤ù♥❣ tä r➡♥❣ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ❝â t❤➸ ✤÷đ❝ t➼♥❤ t♦→♥ t❤ỉ♥❣ q✉❛ ♥❤ú♥❣ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣✳ ❈❤♦ (R, m) ❧➔ ởt tr ữỡ ợ ỹ ✤↕✐ m ✈➔ ❞✐♠R = n ⩾ 1✳ ◆➠♠ ✶✾✼✸✱ ❍♦❝❤st❡r ❬✺❪ ✤➣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t✿ ❈❤♦ x1, , xn ❧➔ ♠ët ❤➺ t❤❛♠ sè ❜➜t ❦ý ❝õ❛ R✱ xt1 xtn ∈ / Rxt+1 + + Rxt+1 ✈ỵ✐ ♠å✐ t n ỵ ([19, 4.1]) x1, , xn ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ R✳ ❑❤✐ ✤â ♣❤➙♥ −n−1 sè s✉② rë♥❣ (x , 1, x , 1) tr♦♥❣ Un+1 R ❧➔ ❦❤→❝ ❦❤ỉ♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ n t ⩾ 0✱ xt1 xtn ∈ / Rxt+1 + + Rxt+1 n , ♥❣❤➽❛ ❧➔ ❤➺ t❤❛♠ sè x1, , xn t❤ä❛ ♠➣♥ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝✳ ❈❤ù♥❣ () tỗ t ởt t s n xt1 xt2 xtn Rxt+1 i , ∈ i=1 t❤❡♦ ●❤✐ ❝❤ó ✷✳✷✳✹✭✸✮✭✐✐✮✱ ❝❤ó♥❣ t❛ ❝â xt1 xtn = 0, t+1 xt+1 , , xn , ✺✾ −n−1 tr♦♥❣ Un+1 R✳ ❚✉② ♥❤✐➯♥✱ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ D = ❞✐❛❣ t+1 D [x1 xn 1]T = xt+1 1 xn xt1 , , xtn , t❤ä❛ ♠➣♥ T −n−1 ✈➔ ✈➻ ✈➟②✱ tr♦♥❣ Un+1 R✱ xt1 xtn = = t+1 t+1 (x1 , , xn , 1) x1 , , x n , (⇐) ●✐↔ sû (x , x n , 1) =0 −n−1 tr♦♥❣ Un+1 R✳ ❑❤✐ ✤â t❤❡♦ ❍➺ q✉↔ ✸✳✶✳✽✱ t❛ ❝â = 0, (x1 , xn , 1) tr♦♥❣ U (x)−n−1 n+1 R✱ ð ✤➙② ❝❤ó♥❣ t❛ ũ tr ú tỗ t α1, , αn ∈ Z+ ✈➔ H′ ∈ Dn+1 (R) s❛♦ ❝❤♦ T ′ H [x1 xn 1] = xα1 xαnn n ✈➔ |H | ∈ T ′ Rxαi i i=1 ✣➦t c = ♠❛① αi | i = 1, , n ✳ ❳➨t ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ❞✐❛❣ xc−α , , xnc−α , 1 , s❛♦ ❝❤♦ ✈➔ t❛ ❝â ♠❛ tr➟♥ H ∈ Dn+1 (R) ✈ỵ✐ H = H′.❞✐❛❣ xc−α , , xc−α n 1 n n n T H [x1 xn 1] = [xc1 xcn T Rxci |H| ∈ 1] , i=1 ❚✉② ♥❤✐➯♥✱ ♥➳✉ D = ❞✐❛❣ n xc−α , , xc−α ,1 n t❤➻ D [x1 xn 1]T = [xc1 xcn 1]T ❙✉② r❛ tø ❇ê ✤➲ ✷✳✶✳✹✱ ♥➳✉ ❦➼ ❤✐➺✉ E ❧➔ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ❞✐❛❣ t❤➻ xc1 , , xcn , 2c |ED| − |EH| ∈ Rx2c + + Rxn , ✈➻ n |EH| = ✈➔ ✈➻ xc1 xcn |H| Rx2c i ∈ i=1 2c |ED| = x2c−1 x2c−1 ∈ Rx2c n + + Rxn P❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ữủ t tú ứ ỵ tt ỡ tự tữỡ ữỡ ợ s n1 ợ ❤➺ t❤❛♠ sè x1, , xn ❝õ❛ R ♣❤➛♥ tû (x , 1, x , 1) tr♦♥❣ Un+1 R ❧➔ n ❦❤→❝ ❦❤æ♥❣✳ ✻✵ ❍➺ q✉↔ ✸✳✷✳✷✳ ([19, 4.3])✳ ❈❤♦ y1, , yn ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ R✳ ❑❤✐ õ tỗ t ởt số tỹ t N s❛♦ ❝❤♦✱ ❦❤✐ h ⩾ t✱ ❤➺ t❤❛♠ sè x1 = y1h, , xn = ynh t❤ä❛ ♠➣♥ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝✳ ∼ n ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❍➺ q✉↔ ✸✳✶✳✽✱ t❛ ❝â U (y)−n−1 n+1 = Hm (R) = tỗ t , , βn ∈ Z+ s❛♦ ❝❤♦ y1β1 , , ynβn , ̸= tr♦♥❣ U (y)−n−1 n+1 R✳ ❉♦ ✤â✱ t❤❡♦ ❍➺ q✉↔ ✸✳✶✳✽✱ y1β1 , , ynβn , ̸= −n−1 tr♦♥❣ Un+1 R✳ ✣➦t t = ♠❛① {β1 , , βn }✳ ❑➳t q ữủ s r tứ ỵ P ự ♠✐♥❤ ✤÷đ❝ ❦➳t t❤ó❝✳ ●❤✐ ❝❤ó ✸✳✷✳✸✳ ✶✳ ❈❤♦ (R, m) ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥ ✤à❛ ♣❤÷ì♥❣ −d−1 ✈ỵ✐ ✐✤➯❛♥ ❝ü❝ ✤↕✐ m ✈➔ ❞✐♠R = d ⩾ 1✳ ❳➨t ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ Ud+1 R ❝õ❛ R tữỡ ự ợ U (R)d+1 x1, , xd ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ R ✈➔ −d−1 R s✐♥❤ ❜ð✐ ♣❤➙♥ sè n1 , , nd ∈ Z+ ✳ ❑❤✐ ✤â ♠æ✤✉♥ ❝♦♥ ①✐❝❧✐❝ ❝õ❛ U (R)d+1 s✉② rë♥❣ xn , , xn , ∈ U (R)−d−1 d+1 R ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✱ tù❝ ❧➔ 1 d d lR R xn1 , , xnd d , < ∞ ❙❤❛r♣ ✈➔ ❍❛♠✐❡❤ ✤÷❛ r❛ ❝➙✉ ❤ä✐ tr♦♥❣ ❬✷✵❪✿ ❈❤♦ x1, , xd ❧➔ ♠ët ❤➺ t❤❛♠ sè R õ tỗ t ởt tự h Q [x1, , xd] s❛♦ ❝❤♦ l R xn1 , , xnd d , = h (n1 , , nd ) ❦❤✐ n1, , nd ≫ 0✱ ✭n1, , nd ợ ọ ữ õ ❝➙✉ tr↔ ❧í✐ ①→❝ ✤à♥❤ ❝❤♦ tr÷í♥❣ ❤đ♣ d ⩾ 3✳ ❚r÷í♥❣ ❤đ♣ d = 1, d = ✤➣ ❝â ❝➙✉ tr↔ ❧í✐ ✭①❡♠ ❬✷✵❪✮✳ ✷✳ ❚r♦♥❣ ❬✷✷❪✱ ❙❤❛r♣ ✈➔ ❩❛❦❡r✐ ✤➣ ✤÷❛ r❛ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ỉ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② rë♥❣✱ ❇✉❝❤s❜❛❝♠ t❤æ♥❣ q✉❛ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣✳ ✻✶ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❬✶✽❪✱ ❬✶✾❪ ❝ư t❤➸ ❧➔✿ ✶✳ ❈❤÷ì♥❣ ✶✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♥❤÷✿ ✣ë ❞➔✐ ♠ỉ✤✉♥✱ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✱ ❈❤✐➲✉ r P trũ tỷ ổ ố ỗ ổ ố ỗ ữỡ ợ t ✷✳ ❈❤÷ì♥❣ ✷✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ✈➲ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ tr♦♥❣ ❬✶✽❪✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ✈➔ ✈➼ ❞ö ✭✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✱ ❱➼ ❞ö ✷✳✶✳✷✮❀ ①➙② ❞ü♥❣ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ M × U ✈ỵ✐ M ❧➔ ♠ët R−♠ỉ✤✉♥ ✈➔ U ❧➔ ♠ët t➟♣ ❝♦♥ t❛♠ ❣✐→❝ ❝õ❛ Rn ✭▼➺♥❤ ✤➲ ✷✳✶✳✺✮❀ ỹ ổ số s rở ỵ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ✈➔ ♠ët sè ✈➼ ❞ö ✭▼➺♥❤ ✤➲ ✷✳✷✳✶✱ ▼➺♥❤ ✤➲ ✷✳✷✳✷✱ ▼➺♥❤ ✤➲ ✷✳✷✳✼✱ ❱➼ ❞ö ✷✳✷✳✸✱ ❱➼ ❞ö ✷✳✷✳✺✱ ❱➼ ❞ư ✷✳✷✳✽✱ ❱➼ ❞ư ✷✳✷✳✾✮✳ ✸✳ ❈❤÷ì♥❣ ✸✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❬✶✾❪ ✈➲ ♠æ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ố ỗ ữỡ ✤➲ ✸✳✶✳✷✱ ❇ê ✤➲ ✸✳✶✳✹✱ ❇ê ✤➲ ✸✳✶✳✺✱ ❍➺ q✉↔ ỵ q trữ tt ỡ tự q số s rở ỵ ✸✳✷✳✶✱ ❍➺ q✉↔ ✸✳✷✳✷✮✳ ✻✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆t✐②❛❤✳▼✳❋ ❛♥❞ ▼❛❝❞♦♥❛❧❞✳■✳●✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✳ ❘❡❛❞✐♥❣ ▼❛ss✱ ✶✾✻✾✳ ❬✷❪ ❇r♦❞♠❛♥♥✳▼✳ ❛♥❞ ❙❤❛r♣✳❘✳❨✳ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝✲ t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❈❛♠❜r✐❞❣❡✱ ✶✾✾✽✳ ❬✸❪ ●♦♣❛❧❛❦r✐s❤♠❡♥✳◆✳❙✳ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✳ ❉✉♠❡✱ ✶✾✺✹✳ ❬✹❪ ●r♦t❤❡♥❞✐❡❝❦✳❆✳ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✹✶✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✶✾✻✼✳ ❬✺❪ ❍♦❝❤st❡r✳▼✳ ❈♦♥tr❛❝t❡❞ ✐❞❡❛❧s ❢r♦♠ ✐♥t❡❣r❛❧ ❡①t❡♥s✐♦♥s ♦❢ r❡❣✉❧❛r r✐♥❣s✳ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✺✶ ✭✶✾✼✸✮✱ ✷✺✲✹✸✳ ❬✻❪ ❍♦❝❤st❡r✳▼✳ ✧❚♦♣✐❝s ✐♥ t❤❡ ❍♦♠♦❧♦❣✐❝❛❧ ❚❤❡♦r② ♦❢ ▼♦❞✉❧❡s ♦✈❡r ❈♦♠♠✉✲ t❛t✐✈❡ ❘✐♥❣s✧✳ ❈♦♥❢❡r❡♥❝❡ ❇♦❛r❞ ♦❢ t❤❡ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s ❘❡❣✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❙❡r✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✷✹✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘✳■✳✱ ✶✾✼✺✳ ❬✼❪ ❍♦❝❤st❡r✳▼✳ ❆ss♦❝✐❛t❡❞ ●r❛❞❡❞ ❘✐♥❣s ❉❡r✐✈❡❞ ❢r♦♠ ■♥t❡❣r❛❧❧② ❈❧♦s❡❞ ■❞❡✲ ❛❧s ❛♥❞ t❤❡ ▲♦❝❛❧ ❍♦♠♦❧♦❣✐❝❛❧ ❈♦♥❥❡❝t✉r❡s✳ Pr❡♣r✐♥t✱ ❯♥✐✈❡rs✐t② ♦❢ ▼✐❝❤✐✲ ❣❛♥✱ ✶✾✽✶✳ ❬✽❪ ❍♦❝❤st❡r✳▼✳ ❈❛♥♦♥✐❝❛❧ ❡❧❡♠❡♥ts ✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❛♥❞ t❤❡ ❞✐✲ r❡❝t s✉♠♠❛♥❞ ❝♦♥❥❡❝t✉r❡✳ ❏✳ ❆❧❣❡❜r❛ ✽✹ ✭✶✾✽✸✮✱ ✺✵✸✲✺✺✸✳ ❬✾❪ ❑❛♣❧❛♥s❦②✳■✳ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣s✳ ❆❧❧②♥ ❛♥❞ ❇❛❝♦♥✳ ❇♦st♦♥✱ ✶✾✼✵✳ ❬✶✵❪ ❑✐r❜②✳❉✳ ❈♦♣r✐♠❛r② ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✳ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✭✷✮ ✻ ✭✶✾✼✸✮✱ ✺✼✶✲✺✼✻✳ ❬✶✶❪ ▼❛❝❞♦♥❛❧❞✳■✳●✳ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣✳ ❙②♠♣♦s✳ ▼❛t❤✳ ✶✶ ✭✶✾✼✸✮✱ ✷✸✲✹✸✳ ❬✶✷❪ ▼❛❝❞♦♥❛❧❞✳■✳● ❛♥❞ ❙❤❛r♣✳❘✳❨✳ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♦❢ ❝❡rt❛✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✭✷✮✱ ✷✸ ✭✶✾✼✷✮✱ ✶✾✼✲✷✵✹✳ ✻✸ ❬✶✸❪ ▼❛ts✉♠✉r❛✳❍✳ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ❚❤❡♦r②✳ ❈❛♠❜r✐❞❣❡ ❙✉t✐❡s ✐♥ ❛❞✈❛♥❝❡❞ ♠❛t❤❡♠❛t✐❝s✱ ◆♦ ✽✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss ✶✾✽✻✳ ❬✶✹❪ ▼❛ts✉♠✉r❛✳❍✳ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✳ ❇❡♥❥❛♠✐♥✳ ❘❡❛❞✐♥❣ ✶✾✽✷✳ ❬✶✺❪ ◆♦rt❤❝♦tt✳❉✳●✳ ■❞❡❛❧ t❤❡♦r②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❈❛♠❜r✐❞❣❡✱ ✶✾✺✸✳ ❬✶✻❪ ◆♦rt❤❝♦tt✳❉✳●✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❍♦♠♦❧♦❣✐❝❛❧ ❆❧❣❡❜r❛✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✳ ❈❛♠❜r✐❞❣❡✱ ✶✾✻✵✳ ❬✶✼❪ ❙❤❛r♣✳❘✳❨✳ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣② t❤❡♦r② ✐♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✭✷✮✱ ✷✶ ✭✶✾✼✵✮✱ ✹✷✺✲✹✸✹✳ ❬✶✽❪ ❙❤❛r♣✳❘✳❨✳ ❛♥❞ ❩❛❦❡r✐✳❍✳ ▼♦❞✉❧❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✳ ▼❛t❤❡♠❛t✐❦❛ ✷✾ ✭✶✾✽✷✮✱ ✸✷✲✹✶✳ ❬✶✾❪ ❙❤❛r♣✳❘✳❨✳ ❛♥❞ ❩❛❦❡r✐✳❍✳ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣② ❛♥❞ ▼♦❞✉❧❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✱ ▼❛t❤❡♠❛t✐❦❛ ✷✾ ✭✶✾✽✷✮✱ ✷✾✻✲✸✵✻✳ ❬✷✵❪ ❙❤❛r♣✳❘✳❨✳ ❛♥❞ ❍❛♠✐❡❤✳◆✳❆✳ ▲❡♥❣t❤s ♦❢ ❝❡rt❛✐♥ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥✳ ❏♦✉r✲ ♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧❣❡❜r❛ ✸✽ ✭✶✾✽✺✮✱ ✸✷✸✲✸✸✻✳ ❬✷✶❪ ❙❤❛r♣✳❘✳❨✳ ❛♥❞ ❩❛❦❡r✐✳❍✳ ●❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s ❛♥❞ ▼♦♥♦♠✐❛❧ ❈♦♥❥❡❝t✉r❡✳ ❏✳❆❧❣❡❜r❛ ✾✷ ✭✶✾✽✺✮✱ ✸✽✵✲✸✽✽✳ ❬✷✷❪ ❙❤❛r♣✳❘✳❨ ❛♥❞ ❍✳❩❛❦❡r✐✱ ●❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✱ ❇✉❝❤s❜❛❝♠ ♠♦❞✉❧❡s ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ▼♦❞✉❧❡s✱ ▼❛t❤✳ ♣r♦❝✳ ❈❛✉❜✳ P❤✐❧❧ ❙♦❝ ✾✽ ✭✶✾✽✺✮ ✹✷✾✲✹✸✻✳ ... ❍➴❈ ◗❯❨ ◆❍❒◆ ✣■◆❍ ❍Ú❯ ❉❯❨ ▼➷✣❯◆ P❍❹◆ ❙➮ ❙❯❨ ❘❐◆● ❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ◆❣➔♥❤✿ ✣❸■ ❙➮ ❱⑨ số ữớ ữợ ◆●❯❨➍◆ ❚❍⑩■ ❍➪❆ ✐ ▼ö❝ ❧ö❝ ▼ö❝ ❧ö❝ ❉❛♥❤ ♠ö❝ ❝→❝ ❦➼ ❤✐➺✉ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥... ▼ët sè t➼♥❤ ❝❤➜t ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ ♣❤➙♥ số s rở ố ỗ ữỡ ❣✐↔ t❤✉②➳t ✤ì♥ t❤ù❝ ✹✽ ✸✳✶ ▼ỉ✤✉♥ ♣❤➙♥ sè s✉② rở ố ỗ ữỡ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷ Ù♥❣... n ❝õ❛ ❤➔♠ tû F ❍➔♠ tỷ ố ỗ ữỡ tự n tữỡ ự ✈ỵ✐ ✐✤➯❛♥ a ❍➔♠ tû ❣✐ỵ✐ ❤↕♥ t❤✉➟♥ ❚➟♣ ❝♦♥ t Rn ợ n số ữỡ ▼ỉ✤✉♥ ♣❤➙♥ sè s✉② rë♥❣ ❝õ❛ R−♠ỉ✤✉♥ M ù♥❣ ✈ỵ✐ t➟♣ ❝♦♥ t❛♠ ❣✐→❝ U ❝õ❛ Rn ❍➔♠ tû tø trũ (R)