$\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$

Abstract

Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a $\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper class $\scr{RS}$