On Automorphism-invariant multiplication modules over a noncommutative ring

Authors : Le Van Thuyet, Truong Cong Quynh

Pages : 73-88

Doi:10.24330/ieja.1411145

View : 103 | Download : 121

Publication Date : 2024-07-12

Article Type : Research Paper

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.
Keywords : Automorphism invariant module, duo ring, quasi duo ring, multiplication module, commutative multiplication of ideals