New generalizations of lifting modules

doi:10.1285/i15900932v36n2p49

New generalizations of lifting modules

T. Amouzegar

Abstract

In this paper, we call a module $M$ almost $\mathcal{I}$ -lifting if, for any element $\phi\in S=End_R(M)$ , there exists a decomposition $r_M\ell_S(\phi)=A\oplus B$ such that $A\subseteq \phi M$ and $\phi M\cap B\ll M$ . This definition generalizes the lifting modules and left generalized semiregular rings. Some properties of these modules are investigated. We show that if $f_1+\cdots + f_n=1$ in $S$ , where $f_i$ ${^{ ,}}$ s are orthogonal central idempotents, then $M$ is an almost $\mathcal{I}$ -lifting module if and only if each $f_iM$ is almost $\mathcal{I}$ -lifting. In addition, we call a module $M$ $\pi$ - $\mathcal{I}$ -lifting if, for any $\phi\in S$ , there exists a decomposition $\phi^nM=eM\oplus N$ for some positive integer $n$ such that $e^2=e\in S$ and $N\ll M$ . We characterize semi- $\pi$ -regular rings in terms of $\pi$ - $\mathcal{I}$ -lifting modules. Moreover, we show that if $M_1$ and $M_2$ are abelian $\pi$ - $\mathcal{I}$ -lifting modules with $Hom_R(M_i, M_j)=0$ for $i\neq j$ , then $M=M_1\oplus M_2$ is a $\pi$ - $\mathcal{I}$ -lifting module.

DOI Code: 10.1285/i15900932v36n2p49

Keywords: Lifting module; $\mathcal{I}$-Lifting module; Semiregular ring; Semi-$\pi$-regular ring

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Note di Matematica

New generalizations of lifting modules

Abstract