On $\tau_M$-Semilocal Modules and Rings

Abstract

Let $\tau_M$ be any preradical for $\sigma[M]$ and $N$ any module in $\sigma[M]$. In [2], Al-Takhman, Lomp and Wisbauer defined and studied the concept of $\tau_M$-supplemented module. In this paper we define the concept of weakly $\tau_M$-supplemented module and investigate some properties of such modules. We show that weakly $\tau_M$-supplemented module $N$ is $\tau_M$-semilocal (i.e., $N/\tau_M(N)$ is semisimple) and that $R$ is a $\tau$-semilocal ring if and only if $_{R}R$ (or $R_R$) is weakly $\tau_M$-supplemented.