Về môđun với epi-dcc

✣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 ❑❍❖❆ ❍➴❈

●✐→♦ ✈✐➯♥ ❤÷î♥❣ ❞➝♥✿

●❙✳ ❚❙✳ ▲➯ ❱➠♥ ❚❤✉②➳t

✣⑨ ◆➂◆● ✕ ✷✵✷✵

▲❮■ ❈❆▼ ✣❖❆◆

❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✳ ❈→❝ sè

❧✐➺✉✱ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷ñ❝ ❛✐ ❝æ♥❣

❜è tr♦♥❣ ❜➜t ❦➻ ❝æ♥❣ tr➻♥❤ ♥➔♦ ❦❤→❝✳

❚→❝ ❣✐↔

P❤❛♥ ❆♥❤ ❚✉➜♥

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INFORMATION PAGE OF MASTER 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 rt 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▼ ❒◆

❱î✐ t➻♥❤ ❝↔♠ ❝❤➙♥ t❤➔♥❤✱ t→❝ ❣✐↔ ①✐♥ ✤÷ñ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥

tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝✱

❑❤♦❛ ❚♦→♥✱ q✉þ t❤➛②✱ ❝æ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❧î♣ ❈❛♦ ❤å❝ ✣↕✐ sè ✈➔ ❧þ t❤✉②➳t

sè ❑✸✺ ✤➣ t➟♥ t➻♥❤ ❤÷î♥❣ ❞➝♥✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ 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✐➳♣

❣✐↔♥❣ ❞↕②✱ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✳ ❱î✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ q✉þ

❜→✉✱ ❚❤➛② ✤➣ ➙♥ ❝➛♥ ❝❤➾ ❜↔♦ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tü t✐♥✱ ✈÷ñt q✉❛ ♥❤ú♥❣ ❦❤â

❦❤➠♥✱ trð ♥❣↕✐ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳

❚→❝ ❣✐↔ ①✐♥ ✤÷ñ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ P●❙✳ ❚❙✳ ❚r÷ì♥❣ ❈æ♥❣ ◗✉ý♥❤

❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ❚❤➛② ✤➣ ❧✉æ♥ t❤❡♦ s→t ❧î♣

❈❛♦ ❤å❝ ✣↕✐ sè ✈➔ ❧þ t❤✉②➳t sè ❑✸✺ ❤÷î♥❣ ❞➝♥ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ ❧î♣ ❝â

✤÷ñ❝ ❦➳t q✉↔ ❤å❝ tèt ♥❤➜t✳

❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧î♣ ❈❛♦ ❤å❝ ✣↕✐ sè ✈➔ ❧þ

t❤✉②➳t sè ❑✸✹✱ ❑✸✺✱ ❑✸✻ ✈➔ ❜↕♥ ❜➧ ♥❣÷í✐ t❤➙♥ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï✱ t↕♦

✤✐➲✉ ❦✐➺♥ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳

❉ò t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣✱ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣

t❤✐➳✉ sât✱ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ❣â♣ þ✱ ❝❤➾ ❞➝♥ ❝õ❛ q✉þ t❤➛②✱ ❝æ ❣✐→♦✱

❝→❝ ❜↕♥ ✤ç♥❣ ♥❣❤✐➺♣ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ q✉❛♥ t➙♠ ✤➳♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉✳

❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦

❚→❝ ❣✐↔

P❤❛♥ ❆♥❤ ❚✉➜♥