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Journals / International Electronic Journal of Algebra / 2024 / Volume: 36 Issue: 36

On Automorphism-invariant multiplication modules over a noncommutative ring

Le Van Thuyet(Hue University)
Truong Cong Quynh(The University of Danang)
Pages
—
Publication Date
2024-07-12
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M_R \cong R$ is injective.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:One of the important classes of modules is the class of multiplication modules over a commutative ring.
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Key Findings & Data:We prove that if $R$ is a right duo ring and $M$ is a multiplication, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M$ is projective.
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Conclusion & Impact:Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M_R \cong R$ is injective.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Automorphism-invariant module · duo ring · quasi-duo ring

Abstract

One of the important classes of modules is the class of multiplication modules over a commutative ring. This topic has been considered by many authors and numerous results have been obtained in this area. After that, Tuganbaev also considered the multiplication module over a noncommutative ring. In this paper, we continue to consider the automorphism-invariance of multiplication modules over a noncommutative ring. We prove that if $R$ is a right duo ring and $M$ is a multiplication, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M$ is projective. Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M_R \cong R$ is injective.

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Cite This Article

Thuyet, L. V., Quynh, T. C. (2024). On Automorphism-invariant multiplication modules over a noncommutative ring. International Electronic Journal of Algebra, Volume: 36 Issue: 36. https://doi.org/10.24330/ieja.1411145

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