New generalizations of lifting modules

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.