Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 5, No 3 (2011)

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On Principally Quasi-Baer Modules

Burcu Ungor, Nazim Agayev, Sait Halicioglu, Abdullah Harmanci


Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$.  In this paper, we introduce a class of modules that is a generalization of principally quasi-Baer rings and Baer modules. The module $_SM$ is called {\it principally quasi-Baer} if for any $m\in M$, $l_S(Sm)=Se$ for some $e^2=e\in S$. We prove that some results of principally quasi-Baer rings  can be extended to principally quasi-Baer modules for this general settings. We prove that if $_SM$ is a regular and semicommutative module, then $_SM$ is principally quasi-Baer. We also prove that if $M_R$ is principally semisimple and $_SM$ is abelian, then $_SM$ is a principally quasi-Baer module. It is shown that any direct summand of a principally quasi-Baer module inherits this property. Among others we show that there is a strong connection between a principally quasi-Baer module $_SM$ and polynomial extension, power series extension, Laurent polynomial extension, Laurent power series extension of $_SM$.

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