✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ P❍❆◆ ❆◆❍ ❚❯❻◆ ❱➋ ▼➷✣❯◆ ❱❰■ ❊P■✕❉❈❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❈❍❯❨➊◆ ◆●⑨◆❍ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ✣⑨ ◆➂◆● ✕ ✷✵✷✵ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ P❍❆◆ ❆◆❍ ❚❯❻◆ ❱➋ ▼➷✣❯◆ ❱❰■ ❊P■✕❉❈❈ ❈❍❯❨➊◆ ◆●⑨◆❍✿ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ▼❶ ❙➮✿ ✻✵✳✹✻✳✵✶✳✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ●✐→♦ ữợ t ✕ ✷✵✷✵ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✳ ❈→❝ sè ❧✐➺✉✱ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❛✐ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦➻ ❝æ♥❣ tr➻♥❤ ♥➔♦ ❦❤→❝✳ ❚→❝ ❣✐↔ P❤❛♥ ❆♥❤ ❚✉➜♥ INFORMATION PAGE OF MAS TER THESIS Name of thesis: On 1nodules with epi-DCC :Major: Algrebra and N u1nber theory Full name of rviaster student: Phai Anh uan Suppervisor: Prof Dr Le Van Thuyet Training institution: The University of Da Nang, University of Ed ucation Abstract: Modular theory has an important role when studying Algebra and there a.re many new issues to be investigated Vie say that a set O of submodules of If satisfies the descending chain condition (often abbreviated as DCC) if in eYery descending cha.in of submodules of r t there exists n EN such that Ln+i = Ln (for all i = 1, 2, ) The family of mod ules satisfying the descending chain condition and its related problems are the basis for studying other issues In a paper by R Dastanpour and A Ghorbani named "Mod ules with epimorphism on chains of submodules", an R-module is said to be satisfied epi-DCC on submodules if in every descending cha.in of submodules of 11, except prob ably a finite number, each module in chain is a homomorphic image of the preceding Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules A semiprime right Goldie ring satisfies epi-DCC on right ide.ls if and only if it is a finite product of full matrix rings over principal right ideal domains Based on this article ) our thesis gives an overview of some results on the properties of modules with epi-DCC, studies other special properties and relationships with related rings Key words: epi-DCC, epi-DCC modules, epi-DCC decreasing sequences, descending cha.in condition, epi-DCC on submodules Student Prof Dr Le Van Thuyet Phan Anh Tuan ▲❮■ ❈❷▼ ❒◆ ❱ỵ✐ t➻♥❤ ❝↔♠ ❝❤➙♥ t❤➔♥❤✱ t→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ P❤á♥❣ t qỵ t ổ ợ số ỵ tt số t t ữợ t ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ●❙✳ ❚❙✳ ▲➯ ❱➠♥ ❚❤✉②➳t✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❍✉➳✱ ♥❣÷í✐ ❚❤➛② trü❝ t✐➳♣ ❣✐↔♥❣ ữợ ợ ỳ tự qỵ ú ✤ï t→❝ ❣✐↔ tü t✐♥✱ ✈÷đt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥✱ trð ♥❣↕✐ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ P●❙✳ ❚❙✳ ❚r÷ì♥❣ ❈ỉ♥❣ ◗✉ý♥❤ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ❚❤➛② ✤➣ ❧✉ỉ♥ t st ợ số ỵ tt số ữợ t ợ ❝â ✤÷đ❝ ❦➳t q✉↔ ❤å❝ tèt ♥❤➜t✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❈❛♦ ❤å❝ ✣↕✐ số ỵ tt số ❜➧ ♥❣÷í✐ t❤➙♥ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❉ò t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣✱ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ qỵ t ổ ỗ ỳ ữớ q t t➔✐ ♥❣❤✐➯♥ ❝ù✉✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❚→❝ ❣✐↔ P❤❛♥ ❆♥❤ ❚✉➜♥ ✐✐✐ ▼Ö❈ ▲Ö❈ ❉❆◆❍ ▼Ö❈ ❈⑩❈ ❑Þ ❍■➏❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ ▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ▼æ✤✉♥ tü ❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳ ▼æ✤✉♥ ♥ë✐ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✻ ✶✳✹✳ ▼ỉ✤✉♥ ✤ì♥✱ ♠ỉ✤✉♥ ♥û❛ ✤ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✺✳ ❈➠♥ ✈➔ ✤➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✻✳ ❱➔♥❤ ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✼ ❈❍×❒◆● ✷✳ ✣■➋❯ ❑■➏◆ ❉❈❈ ❱⑨ ❊P■✲❉❈❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳ ✣✐➲✉ ❦✐➺♥ ❞➣② ❣✐↔♠ ✭❉❈❈✮✭♠æ✤✉♥ ✈➔ ✈➔♥❤ ❆rt✐♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳ ▼ỉ✤✉♥ ✈ỵ✐ t♦➔♥ ❝➜✉ ❡♣✐✲❝♦ tr➯♥ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ✳ ✳ ✳ ✷✼ ✷✳✸✳ ❱➔♥❤ ●♦❧❞✐❡ ♣❤↔✐ ♥û❛ ♥❣✉②➯♥ tè ✈ỵ✐ ❡♣✐✲❝♦ tr➯♥ ❞➣② ✐✤➯❛♥ ♣❤↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹✳ ❱➔♥❤ tr♦♥❣ ✤â ♠å✐ ♠æ✤✉♥ t❤ä❛ ♠➣♥ ❡♣✐✲❝♦ tr➯♥ ❝→❝ ❞➣② ❣✐↔♠ ✸✽ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✐✈ ❉❆◆❍ ▼Ư❈ ❈⑩❈ ❑Þ ❍■➏❯ ∼ = N N M ess M A⊕B n Mi i=1 R MR Rad(M ) J(R) Soc(M ) End(M ) HomR (A, B) ❛♥♥R(x) ✣➥♥❣ ❝➜✉ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ M N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝èt ②➳✉ ❝õ❛ ♠æ✤✉♥ M ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ❤❛✐ ♠æ✤✉♥ A ✈➔ B ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♠æ✤✉♥ Mi, ≤ i ≤ n ❱➔♥❤ ❝â ✤ì♥ ✈à = ▼æ✤✉♥ ♣❤↔✐ tr➯♥ ✈➔♥❤ R ❈➠♥ ❏❛❝♦❜s♦♥ ❝õ❛ ♠æ✤✉♥ M ❈➠♥ ❏❛❝♦❜s♦♥ ❝õ❛ ✈➔♥❤ R ✣➳ ❝õ❛ ♠æ✤✉♥ M ❱➔♥❤ tỹ ỗ M ổ ỗ ❣✐ú❛ ❝→❝ R✲♠æ✤✉♥ A ✈➔ B ▲✐♥❤ tû ❤â❛ ❝õ❛ x tr♦♥❣ R ❑➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ✶ ▼Ð ✣❺❯ ỵ t ỵ tt ổ õ ✈❛✐ trá q✉❛♥ trå♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✣↕✐ sè ❦➳t ủ ỏ ợ ữủ q t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❳➨t ✤➳♥ t➟♣ ❤đ♣ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ ♠ët ♠ỉ✤✉♥ ❝â ❝ị♥❣ t➼♥❤ ❝❤➜t ❤❛② t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤ú✉ ❤↕♥ ♥➔♦ ✤â ✤➸ ❝❤ó♥❣ t❛ s➢♣ ①➳♣ ❝❤ó♥❣✱ ♣❤➙♥ ❧♦↕✐ ❝❤ó♥❣ ❧➔ ♠ët ✈➜♥ ✤➲ t❤÷í♥❣ ❣➦♣ ♣❤↔✐ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♠ỉ✤✉♥✳ ❈❤ó♥❣ t❛ ✤➣ ❜✐➳t✱ t➟♣ Ω ❝→❝ ♠æ✤✉♥ ❝♦♥ ♥➔♦ ✤â ❝õ❛ M ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❞➣② tữớ ữủ t tt tr trữớ ủ ợ ♠å✐ ❞➣② L1 ≥ L2 ≥ ≥ Ln tr tỗ t nN ✤➸ ❝❤♦ Ln+i = Ln ✭✈ỵ✐ ♠å✐ i = 1, 2, ✮✳ ▲ỵ♣ ❝→❝ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❞➣② ❣✐↔♠ ✭❉❈❈✮ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ tỵ✐ ♥â ❧➔ ♥➲♥ t↔♥❣ ✤➸ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ✈➜♥ ✤➲ ❦❤→❝✳ ▼ỉ✤✉♥ ♥❤÷ ✈➟② ✤÷đ❝ ❣å✐ ❧➔ ❆rt✐♥ ✈➔ ✈➔♥❤ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❆rt✐♥ ♣❤↔✐ ✈➔ tr→✐✳ ▲ỵ♣ ✈➔♥❤ ❆rt✐♥ ♥➔② ✤â♥❣ ✈❛✐ trá q trồ tr ỵ tt ởt ỷ ✤ì♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ✈➔♥❤ ❆rt✐♥ ♣❤↔✐ ❤❛② tr→✐ ❝ị♥❣ ✈ỵ✐ ❝➠♥ ❏❛❝♦❜s♦♥ ❜➡♥❣ ✵✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ❝õ❛ ❘✳ ❉❛st❛♥♣♦✉r ❛♥❞ ❆✳ ●❤♦r❜❛♥✐ ✏ ▼♦❞✉❧❡s ✇✐t❤ ❡♣✐♠♦r♣❤✐s♠ ♦♥ ❝❤❛✐♥s ♦❢ s✉❜♠♦❞✉❧❡s ✑✱ ♠ët R✲♠æ✤✉♥ M t❤ä❛ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ♥➳✉ tr♦♥❣ ♠å✐ ❞➣② ❣✐↔♠ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M✱ trø ♠ët sè ❤ú✉ ❤↕♥✱ ộ ổ tr ởt ỗ ♠æ✤✉♥ ❦➳ t✐➳♣✳ ▲✐➺✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ❝â ✤ó♥❣ ✈ỵ✐ ♠ỉ✤✉♥ ✈ỵ✐ ❡♣✐✕❉❈❈ ❤❛② ❦❤ỉ♥❣❄ ◆❣♦➔✐ r❛✱ ♥â ❝á♥ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ♥➔♦ ❦❤→❝❄ ▼è✐ ợ q õ ữ t❤➳ ♥➔♦❄ ◆❤➡♠ t➻♠ ❤✐➸✉ ✈➲ ♥❤ú♥❣ ✈➜♥ ✤➲ ♥➔②✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤ ❧➔ ✏ ❱➋ ▼➷✣❯◆ ❱❰■ ❡♣✐✕❉❈❈ ✑✳ ✷✳ ▼ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ✷ + ◆❣❤✐➯♥ ❝ù✉ ♠ỉ✤✉♥ ✈ỵ✐ ❉❈❈ ✈➔ ❡♣✐✕❉❈❈ ❝ị♥❣ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✳ + ❉ü❛ ❝❤➼♥❤ tr➯♥ ❜➔✐ ❜→♦ ❝õ❛ ❘✳ ❉❛st❛♥♣♦✉r ❛♥❞ ❆✳ ●❤♦r❜❛♥✐ ✏ ✇✐t❤ ❡♣✐♠♦r♣❤✐s♠ ♦♥ ❝❤❛✐♥s ♦❢ s✉❜♠♦❞✉❧❡s s ú tổ trữợ t tờ q ❦➳t q✉↔ tø ❜➔✐ ❜→♦ ♥➔② ✈➔ ❝→❝ ❜➔✐ ❜→♦ ❦❤→❝✱ tr♦♥❣ s→❝❤ ❜➡♥❣ ❝→❝❤ ❧➔♠ t÷í♥❣ ♠✐♥❤ ❝→❝ ❝❤ù♥❣ ♠✐♥❤✱ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣✳ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ + ◆❣❤✐➯♥ ❝ù✉ ✈➲ ♠ỉ✤✉♥ ✈➔ rt ỵ t t ❝→❝ ✈➼ ❞ư✳ + ◆❣❤✐➯♥ ❝ù✉ ✈➲ ♠ỉ✤✉♥ ✈➔ ✈➔♥❤ ✈ỵ✐ ❡♣✐✕❉❈❈✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ + ❉ü❛ tr➯♥ ❝ì sð ✤➣ ❜✐➳t ✈➲ ♠ỉ✤✉♥✱ ✈➲ ✈➔♥❤ ❆rt✐♥✱ ✳✳✳ ❝ị♥❣ ✈ỵ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠æ✤✉♥ ✈➔ ✈➔♥❤ ❆rt✐♥✱ ♠ỉ✤✉♥ ✈➔ ✈➔♥❤ ✈ỵ✐ ❡♣✐✕❉❈❈✳ +❚r❛♦ ✤ê✐✱ t❤↔♦ ợ ữớ ữợ r q õ ✺✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ❇è ❝ư❝ ❝õ❛ ❧✉➟♥ ✈➠♥ ỗ ử t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠ỉ✤✉♥ ✤➸ ❧➔♠ ❝ì sð ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♠ỉ✤✉♥✱ ♠ỉ✤✉♥ tü ❞♦✱ ♠ỉ✤✉♥ ♥ë✐ ①↕✱ ♠ỉ✤✉♥ ✤ì♥✱ ♠ỉ✤✉♥ ♥û❛ ✤ì♥✱ ❝➠♥ ✈➔ ✤➳✱ ✈➔♥❤ ❝❤➼♥❤ q✉✐✱ ✈➔♥❤ ●♦❧❞✐❡✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ✤✐➲✉ ❦✐➺♥ ❞➣② ❣✐↔♠ ✭❉❈❈✮ ✤è✐ ✈ỵ✐ ❝→❝ ♠ỉ✤✉♥ ❝♦♥✱ ✤â ❝❤➼♥❤ ❧➔ ♠æ✤✉♥ ✈➔ ✈➔♥❤ ❆rt✐♥✱ ♠æ✤✉♥ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❡♣✐✕ ❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✱ t➻♠ ❤✐➸✉ ✈➔ ✤÷❛ r❛ ♠ët sè ❦➳t q✉↔ ✈➲ ❡♣✐✕❉❈❈ ♣❤↔✐ ✤è✐ ✈ỵ✐ ✈➔♥❤ ●♦❧❞✐❡ ♣❤↔✐ ♥û❛ ♥❣✉②➯♥ tè ✈➔ t➻♠ ❤✐➸✉✱ tr➻♥❤ ❜➔② ✈➲ ✈➔♥❤ ♠➔ tr♦♥❣ ✤â t➜t ❝↔ ❝→❝ ♠æ✤✉♥ t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ✹✵ ∞ i=1 E(P )✳ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❱➻ ∞ i=1 E(P ) ❝â ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ♥➯♥ tỗ t k N tỗ t t ∞ ∞ E(P ) → P ⊕ ϕk : i=k E(P ) i=k+1 ◆❤÷♥❣ P ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ tỗ t ởt t k : E(P )(n) → P ✈ỵ✐ n ∈ N✳ ❚ø t➼♥❤ ①↕ ↔♥❤ ❝õ❛ P ♥➯♥ ψk t→❝❤ ✤÷đ❝ ✈➔ P ✤➥♥❣ ❝➜✉ ✈ỵ✐ tê♥❣ trü❝ t✐➳♣ ❝→❝ sè ❤↕♥❣ ❝õ❛ E(P )(n) ✳ ❉♦ ✤â✱ P ❧➔ ♥ë✐ ①↕✳ ▼➺♥❤ ✤➲ ✷✳✹✳✹✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥✳ ◆➳✉ M (N) t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ t❤➻ M (N) ❧➔ ❡♣✐✲❝♦✳ ❈❤ù♥❣ L = M (N) ú ỵ r L(N) ∼ = L t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❈❤♦ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ L✳ ❳➨t ❞➣② ∞ ∞ L≥N⊕ L i=2 ( N) i=1 ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ L ∞ ∞ L≥L⊕ ≥ ✳ ❱➻ L L ≥ i=3 i=2 ( N) ❝â tr ổ tỗ t k N tỗ t ởt t LN k : i=k L i=k+1 tỗ t t ❝➜✉ ∞ ∞ L ( N) L→N⊕ → i=k L → N i=k+1 ❉♦ ✤â✱ L ❧➔ ❡♣✐✲❝♦✳ ❚❤❡♦ ❬✶✺❪✱ t❛ ❣å✐ R✲♠ỉ✤✉♥ M ❧✐➯♥ tư❝ ♥➳✉ M t❤ä❛ ♠➣♥ ữợ ổ M ❧➔ ❝èt ②➳✉ tr♦♥❣ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ M ✳ ❈✷✿ ▼å✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ M ❝ơ♥❣ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ M ✳ ✹✶ ❱➔♥❤ R ❧➔ ❧✐➯♥ tö❝ ♣❤↔✐ ♥➳✉ RR ❧➔ ❧✐➯♥ tö❝✳ ❘ã r➔♥❣✱ ♠ỉ✤✉♥ ♥ë✐ ①↕ ❧➔ ❧✐➯♥ tư❝✳ ❱➔♥❤ tü ♥ë✐ ①↕ ♣❤↔✐ ❧➔ ❧✐➯♥ tư❝ ♣❤↔✐✳ ❱ỵ✐ ✈➔♥❤ ❧✐➯♥ tư❝ ♣❤↔✐✱ t❛ ❝â t➼♥❤ ❝❤➜t s❛✉ ✤➙②✿ ❇ê ✤➲ ✷✳✹✳✺✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❧✐➯♥ tö❝ ♣❤↔✐ s❛♦ ❝❤♦ ♠å✐ ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ R ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✤➳♠ ✤÷đ❝✳ ❑❤✐ ✤â✱ R ❧➔ ✈➔♥❤ ♥û❛ ❤♦➔♥ ❝❤➾♥❤✳ ▼➺♥❤ ✤➲ ✷✳✹✳✻✳ ❈❤♦ R ❧➔ ✈➔♥❤ s❛♦ ❝❤♦ E(RR )(N) t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❑❤✐ ✤â✱ R ❧➔ ✈➔♥❤ ♥û❛ ❤♦➔♥ ❝❤➾♥❤✱ tü ♥ë✐ ①↕ ♣❤↔✐✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✳✸✱ R ❧➔ tü ♥ë✐ ①↕ ♣❤↔✐✳ ❱➻ E(RR )(N) t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ♥➯♥ R(N) ❝ô♥❣ t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈✳ ❉♦ R(N) ❧➔ ❡♣✐✲❝♦ ✭t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✳✹✮✳ ❈❤♦ I ❧➔ ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ R✳ ❳➨t I ố ữ ổ R(N) tỗ t ởt ỗ tứ R(N) I õ I ❧➔ ✤➳♠ ✤÷đ❝ s✐♥❤✳ ❚ø ❇ê ✤➲ ✷✳✹✳✺✱ R ❧➔ ♥û❛ ❤♦➔♥ ❝❤➾♥❤✳ ▼➺♥❤ ✤➲ ✷✳✹✳✼✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❤♦➔♥ ❝❤➾♥❤ ♣❤↔✐✱ tü ♥ë✐ ①↕ ♣❤↔✐ ✈ỵ✐ ❡♣✐✕ ❉❈❈ ♣❤↔✐✳ ❑❤✐ ✤â✱ J(R) ❧➔ ❧ô② ❧✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t J = J(R)✳ ❑❤✐ ✤â✱ J ❧➔ ❚✲❧ô② ❧✐♥❤ ♣❤↔✐✳ ●å✐ e ❧➔ ❧ơ② ✤➥♥❣ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ R✳ ❑❤✐ õ eRR ổ ữỡ ợ ổ ỹ t eJ s ự tỗ t↕✐ m ∈ N s❛♦ ❝❤♦ eJ m = 0✳ ❘ã r➔♥❣✱ eRR ❝â t➼♥❤ ❝❤➜t ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❳➨t ❞➣② eR ≥ eJ ≥ eJ ≥ õ tỗ t k N s ộ i k tỗ t ♠ët t♦➔♥ ❝➜✉ ϕi : eJ i → eJ i+1 ◆❤÷♥❣ eRR ❧➔ ♥ë✐ ①↕ ✈➔ EndR (eR) ∼ = eRe ữ ợ i k ✱ ϕi ♠ð rë♥❣ t❤➔♥❤ ♠ët t♦➔♥ ❝➜✉ ♥➔♦ ✤â eRR õ tỗ t xi R s❛♦ ❝❤♦ exi eRJ i = eJ i+1 ✹✷ ●✐↔ sû r➡♥❣ eJ m = ✈ỵ✐ ♠å✐ m ∈ N✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ i ≥ k ✱ exi e ∈ J ✳ ❚❤➟t ✈➟②✱ ❧➜② i ≥ k ✈➔ ❣✐↔ sû r➡♥❣ exi eR = eR ❑❤✐ ✤â✱ eJ i+1 = exi e(eJ i ) = exi eRJ i = eRJ i = eJ i ❚❤❡♦ ❇ê ✤➲ ✷✳✹✳✷✱ eJ i = 0✱ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â✱ exieR eR ✈➔ exieR ≤ eJ ≤ J ✳ ❉♦ ✤â✱ exi e ∈ J ✳ ❇➙② ❣✐í✱ ✈➻ = eJ k+1 = exk e(k ) ♥➯♥ exk e = 0✳ ✣✐➲✉ ♥➔② ❝ô♥❣ ❝❤ù♥❣ tä r➡♥❣ = eJ k+2 = exk+1 e(eJ k+1 ) = (exk+1 e)(exk e)(eJ k ) ✈➻ (exk+1 e)(exk e) = ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ ♥➔②✱ t❛ ✤✐ ✤➳♥ ❦➳t ❧✉➟♥ (exk+m e) (exk+1 e)(exk e) = ✈ỵ✐ ♠å✐ m ∈ N✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❦❤➥♥❣ ✤à♥❤ J ❧➔ ❚✲❧ơ② ❧✐♥❤ ♣❤↔✐✳ ❉♦ ✤â✱ eJ m = ✈ỵ✐ m ♥➔♦ ✤â t❤✉ë❝ N✳ R ỷ tỗ t ❧ơ② ✤➥♥❣ ✤à❛ ♣❤÷ì♥❣ trü❝ ❣✐❛♦ rí✐ ♥❤❛✉ e1, e2, , en tr♦♥❣ R s❛♦ ❝❤♦ e1 + e2 + · · · + en = ỳ ỵ tr ợ ộ i {1, 2, , n} tỗ t mi ∈ N s❛♦ ❝❤♦ eiJ m = 0✳ ✣➦t i m = max{m1 , m2 , , mn } ❑❤✐ ✤â✱ eiJ m = ✈ỵ✐ ♠å✐ i ∈ {1, 2, , n} ❉♦ ✤â✱ J m = (e1 + · · · + en)J m = ỵ R ✈➔♥❤ s❛♦ ❝❤♦ ♠å✐ R✲♠æ✤✉♥ ♣❤↔✐ t❤ä❛ ♠➣♥ ❡♣✐✕ ❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❑❤✐ ✤â✱ R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ❆rt✐♥✳ ✹✸ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû R ❧➔ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â✱ ♠å✐ R✲♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♠å✐ R✲♠æ✤✉♥ ❧➔ ❝♦✳ ●å✐ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ❑❤✐ ✤â✱ M (N) ❝â t➼♥❤ ❝❤➜t ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✹✳✹ s✉② r❛ M (N) ❧➔ ❡♣✐✲❝♦✳ ●å✐ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ✵ ❝õ❛ M ✳ ❳➨t N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M (N) t❤➻ tỗ t ởt t : M ( N) N õ ởt ỗ tø M ❧➯♥ N ✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä M rtrtt ữ t ỵ ♠➔ ♠å✐ ♠æ✤✉♥ ❧➔ ❝♦ ❧➔ ✈➔♥❤ ❝ü❝ ✤↕✐ ♣❤↔✐ ✭♠å✐ ♠æ✤✉♥ ❝â ♠æ✤✉♥ ❝♦♥ ❝ü❝ ✤↕✐✮✳ ❚ø ✤â s✉② r❛ R ❧➔ ✈➔♥❤ ❝ü❝ ✤↕✐ ♣❤↔✐✳ ❚✐➳♣ t❤❡♦✱ tø ▼➺♥❤ ✤➲ ✷✳✹✳✻ t❛ ❝â R ❧➔ ♠ët ✈➔♥❤ ♥û❛ ❤♦➔♥ ❝❤➾♥❤ tü ♥ë✐ ①↕ ♣❤↔✐✳ ❉♦ ✤â✱ R/J(R) ❧➔ ❆rt✐♥✐❛♥ ♥û❛ ✤ì♥✳ ❱➻ R ❧➔ ✈➔♥❤ ❝ü❝ ✤↕✐ ♣❤↔✐ ♥➯♥ s✉② r❛ R ❧➔ ✈➔♥❤ ❤♦➔♥ ❝❤➾♥❤ ♣❤↔✐✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✳✼ t❤➻ J(R) ❧➔ ❧ô② ❧✐♥❤ ♥➯♥ R ❝ơ♥❣ ❧➔ ♥û❛ ❤♦➔♥ ❝❤➾♥❤ tr→✐✳ ❚÷ì♥❣ tü✱ R/J(R)2 ❧➔ ♠ët ✈➔♥❤ ❤♦➔♥ ❝❤➾♥❤ tr→✐ ✈➔ tü ♥ë✐ ①↕ ♣❤↔✐✳ ❚❤❡♦ ❬✻✱ ❍➺ q✉↔ ✷✳✺❪✱ ♠ët ✈➔♥❤ ♥û❛ ✤à❛ ♣❤÷ì♥❣ tỹ ợ r s ữỡ tü❛ ❋r♦❜❡♥✐✉s✳ ❉♦ ✤â✱ R/J(R)2 ❧➔ tü❛ ❋r♦❜❡♥✐✉s✳ ❉♦ ✤â✱ J(R)/J(R)2 ❧➔ R/J(R)2 ✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tø ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❚❤❡♦ ❬✶✻✱ ❇ê ✤➲ ✶✶❪✱ ✈➔♥❤ ❤♦➔♥ ❝❤➾♥❤ tr→✐ tü ♥ë✐ ①↕ ♣❤↔✐ S tr♦♥❣ ✤â J(S)/J(S)2 ❧➔ S ✲♠æ✤✉♥ ♣❤↔✐ ❤ú✉ ❤↕♥ s✐♥❤ ❧➔ tü❛ ❋r♦❜❡♥✐✉s✳ ❉♦ ✤â✱ R ❧➔ tü❛ ❋r♦❜❡♥✐✉s✳ ❙✉② r❛ R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ❆rt✐♥✳ ✹✹ ❑➌❚ ▲❯❾◆ ❉ü❛ ✈➔♦ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ t→❝ ❣✐↔ ✤➣ ♥❣❤✐➯♥ ❝ù✉ t➻♠ ❤✐➸✉ ✈➔ tê♥❣ ❤đ♣ ✤➸ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ rữợ t tờ ủ ổ rt ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â✳ ✣➦❝ ❜✐➺t tr♦♥❣ ú tổ tr ỵ t➼❝❤ ♠ỉ✤✉♥ ♥ë✐ ①↕ tr➯♥ ✈➔♥❤ ❆rt✐♥✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ✈➔ t❤➸ ❤✐➺♥ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ✈ỵ✐ ❝❤ó♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✽ s✉② r❛ sỹ tỗ t ổ t tr ổ ổ s✉② ❜✐➳♥ ❦❤→❝ ✵ ✈ỵ✐ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠ỉ✤✉♥ ❝♦♥✳ ❈❤ó♥❣ tỉ✐ t✐➳♣ tư❝ sû ❞ư♥❣ ♠ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣ ✈➲ ✈➔♥❤ ✤➸ t➻♠ ❤✐➸✉ ✈➲ ✈➔♥❤ ●♦❧❞✐❡ ♣❤↔✐ ♥û❛ ♥❣✉②➯♥ tè ✈ỵ✐ ❡♣✐✲❝♦ tr➯♥ ❞➣② ✐✤➯❛♥ ♣❤↔✐✳ ỵ t R ✈➔♥❤ ●♦❧❞✐❡ ♥û❛ ♥❣✉②➯♥ tè t❤ä❛ ♠➣♥ ❡♣✐✲❉❈❈ ❦❤✐ R ✤➥♥❣ ❝➜✉ ✈ỵ✐ t➼❝❤ ❤ú✉ ❤↕♥ ❝→❝ ✈➔♥❤ ♠❛ tr➟♥ ✤➛② ✤õ tr➯♥ ♠✐➲♥ ♥❣✉②➯♥ ✐✤➯❛♥ ♣❤↔✐ ❝❤➼♥❤ t❤➻ R ❧➔ ♠ët ✈➔♥❤ ✐✤➯❛♥ ♣❤↔✐ ❝❤➼♥❤✳ ❚r➻♥❤ ❜➔② ✈➲ ✈➔♥❤ ♠➔ tr♦♥❣ ✤â t➜t ❝↔ ❝→❝ ♠æ✤✉♥ t❤ä❛ ♠➣♥ ❡♣✐✕❉❈❈ tr➯♥ ❝→❝ ♠ỉ✤✉♥ ❝♦♥✳ ❙û ❞ư♥❣ ♠ët sè ❦➳t q✉↔ t ỵ t ♠è✐ q✉❛♥ ❤➺ ❝õ❛ ✈➔♥❤ R ❝â ♠å✐ R✲♠æ✤✉♥ ♣❤↔✐ t❤ä❛ ♠➣♥ ❡♣✐✲❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ t❤➻ ❝❤➼♥❤ ❧➔ rt sỹ tờ ủ ỗ tớ tt õ ự ỵ ✤➲✱ ❤➺ q✉↔ ✈➔ t❤➸ ❤✐➺♥ ♠ët sè ✈➼ ❞ö ✈➲ ♠ỉ✤✉♥ ✈ỵ✐ ❡♣✐✲❉❈❈✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ tỉ✐ s➩ t tử t ữủ ỵ ✷✳✹✳✽ r➡♥❣ ♥➳✉ R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ❆rt✐♥ t❤➻ ♠å✐ R✲♠æ✤✉♥ ♣❤↔✐ ❝â t❤ä❛ ♠➣♥ ❡♣✐✲❉❈❈ tr➯♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❤❛② ❦❤æ♥❣❄ tæ✐ s➩ t➻♠ ❤✐➸✉ t❤➯♠ ❝→❝ t➼♥❤ ❝❤➜t ❦❤→❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠æ✤✉♥ ✈➔ ✈➔♥❤ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❡♣✐✲❉❈❈✳ ❚→❝ ❣✐↔ ✹✺ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ t ỵ tt ổ ◗✉ý♥❤✱ ▲✳ ❱✳ ❚❤✉②➳t ✭✷✵✶✸✮✱ ◆❳❇ ✣↕✐ ❤å❝ ❍✉➳✳ ❬✷❪ ▲✳ ❱✳ ❚❤✉②➳t✱ ▲✳ ✣✳ ❚❤♦❛♥❣ ✭✷✵✶✼✮✱ ❱➔♥❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❤ú✉ ❤↕♥✱ ◆❳❇ ✣↕✐ ❤å❝ ❍✉➳✳ ❬✸❪ ❚✳ ❈✳ ◗✉ý♥❤✱ ▲✳ ❱✳ ❚❤✉②➳t ✭✷✵✶✾✮✱ ▼æ✤✉♥ ✈➔ ✈➔♥❤✱ ◆❳❇ ✣↕✐ ❤å❝ ❍✉➳✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❋✳ ❲✳ ❆♥❞❡rs♦♥✱ ❑✳ ❘✳ ❋✉❧❧❡r ✭✶✾✾✷✮✱ ❘✐♥❣s ❛♥❞ ❈❛t❡❣♦r✐❡s ♦❢ ▼♦❞✉❧❡s✱ ◆❡✇ ❨♦r❦✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✳ ❬✺❪ ❘✳ ❉❛st❛♥♣♦✉r ❛♥❞ ❆✳ ●❤♦r❜❛♥✐ ✭✷✵✶✼✮✱ ✏▼♦❞✉❧❡s ✇✐t❤ ❡♣✐♠♦r♣❤✐s♠ ♦♥ ❝❤❛✐♥s ♦❢ s✉❜♠♦❞✉❧❡s✑✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥✱ ❱♦❧✳ ✶✻✱ ◆♦✳ ✻✱ ✶✽ ♣❛❣❡s✳ ❬✻❪ ❈✳ ❋❛✐t❤ ❛♥❞ ❉✳ ❱✳ ❍✉②♥❤ ✭✷✵✵✷✮✱ ✏❲❤❡♥ s❡❧❢✲✐♥❥❡❝t✐✈❡ r✐♥❣s ❛r❡ ◗❋✿ ❆ r❡♣♦rt ♦♥ ❛ ♣r♦❜❧❡♠✑✱ ❏✳ ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ❱♦❧✳ ✶ ✭✶✮✱ ♣♣✳ ✼✺✲✶✵✺✳ ❬✼❪ ❆✳ ●❤♦r❜❛♥✐ ❛♥❞ ▼✳ ❘✳ ❱❡❞❛❞✐ ✭✷✵✵✾✮✱ ✏❊♣✐✲r❡tr❛❝t❛❜❧❡ ♠♦❞✉❧❡s ❛♥❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s✑✱ ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ■r❛♥✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✸✺ ◆♦✳ ✶✱ ♣♣ ✶✺✺✲✶✻✻✳ ❬✽❪ ❑✳ ❘✳ ●♦♦❞❡❛r❧ ✭✶✾✼✻✮✱ ✏❘✐♥❣ ❚❤❡♦r②✿ ◆♦♥s✐♥❣✉❧❛r ❘✐♥❣s ❛♥❞ ▼♦❞✲ ✉❧❡s✑✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✸✸✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✳ ❬✾❪ ❑✳ ❘✳ ●♦♦❞❡❛r❧ ❛♥❞ ❘✳ ❇✳ ❲❛r❢✐❡❧❞ ✭✷✵✵✹✮✱ ❝♦♠♠✉t❛t✐✈❡ ◆♦❡t❤❡r✐❛♥ ❘✐♥❣s✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ◆♦♥✲ ✷♥❞ ❡❞♥✳✱ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦✲ ❝✐❡t② ❙t✉❞❡♥t ❚❡①ts✱ ❱♦❧✳ ✻✶✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ✹✻ ❬✶✵❪ ▼✳ ❚✳ ❑♦s❛♥ ❛♥❞ ❏✳ ❩❡♠❧✐❝❦❛ ✭✷✵✶✹✮✱ ✏▼♦❞✲r❡tr❛❝t❛❜❧❡ r✐♥❣s✑✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ❱♦❧✳ ✹✷ ✭✸✮✱ ♣♣✳ ✾✾✽✲✶✵✶✵✳ ❬✶✶❪ ❚✳ ❨✳ ▲❛♠ ✭✶✾✾✾✮✱ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❘✐♥❣s✱ ●r❛❞✉✲ ❛t❡ ❚❡①ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✸✶✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ❬✶✷❪ ❚✳ ❨✳ ▲❛♠ ✭✶✾✾✽✮✱ ▲❡❝t✉r❡s ♦♥ ▼♦❞✉❧❡s ❛♥❞ ❘✐♥❣s✱ ●r❛❞✉❛t❡ ❚❡①ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✽✾✱ ◆❡✇ ❨♦r❦✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✳ ❬✶✸❪ ❏✳ ❈✳ ▼❝❈♦♥♥❡❧❧ ❛♥❞ ❏✳ ❈✳ ❘♦❜s♦♥ ✭✶✾✽✼✮✱ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ◆♦❡t❤❡✲ r✐❛♥ ❘✐♥❣s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❏♦❤♥ ❲✐❧❡② ✫ ❙♦♥s✱ ❈❤✐❝❤❡st❡r✳ ❬✶✹❪ ❍✳ ▼♦st❛❢❛♥❛s❛❜ ✭✷✵✶✸✮✱ ✏❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❡♣✐✲r❡tr❛❝t❛❜❧❡ ❛♥❞ ❝♦✲❡♣✐✲ r❡tr❛❝t❛❜❧❡ ♠♦❞✉❧❡s✑✱ ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ■r❛♥✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✸✾ ◆♦✳ ✶✱ ♣♣ ✾✵✸✲✾✶✼✳ ❬✶✺❪ ❲✳ ❑✳ ◆✐❝❤♦❧s♦♥ ❛♥❞ ▼✳ ❋✳ ❨♦✉s✐❢ ✭✷✵✵✸✮✱ ◗✉❛s✐✲❋r♦❜❡♥✐✉s ❘✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❚r❛❝ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✺✽✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✶✻❪ ❇✳ ▲✳ ❖s♦❢s❦② ✭✶✾✻✻✮✱ ✏❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ q✉❛s✐✲❋r♦❜❡♥✐✉s r✐♥❣s✑✱ ❏✳ ❆❧❣❡❜r❛✱ ❱♦❧✳ ✹✱ ♣♣✳ ✸✼✸✕✸✽✼✳ ❬✶✼❪ ❑✳ ❱❛r❛❞❛r❛❥❛♥ ✭✷✵✵✽✮✱ ✏❆♥t✐ ❍♦♣❢✐❛♥ ❛♥❞ ❛♥t✐ ❝♦✲❍♦♣❢✐❛♥ ♠♦❞✉❧❡s✑✱ ❆▼❙ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ❙❡r✳✱ ❱♦❧✳ ✹✺✻✱ ♣♣✳ ✷✵✺✕✷✶✽✳ ... modules with epi- DCC, studies other special properties and relationships with related rings Key words: epi- DCC, epi- DCC modules, epi- DCC decreasing sequences, descending cha.in condition, epi- DCC on... paper by R Dastanpour and A Ghorbani named "Mod ules with epimorphism on chains of submodules", an R-module is said to be satisfied epi- DCC on submodules if in every descending cha.in of submodules... principal ideal domains are examples of such modules A semiprime right Goldie ring satisfies epi- DCC on right ide.ls if and only if it is a finite product of full matrix rings over principal