${\mathcal{Z}}^{\ast }$-Semilocal Modules and the Proper Class $\mathrm{\mathcal{R}}\mathcal{S}$

Over an arbitrary ring, a module M is said to be ${\mathcal{Z}}^{\ast }$-semilocal if every submodule U of M has a ${\mathcal{Z}}^{\ast }$ -supplement V in M, i.e., M = U + V and $U\cap \kern0.5em V\subseteq {\mathcal{Z}}^{\ast }(V),$ where ${\mathcal{Z}}^{\ast }(V)=\left\{m\in \left.V\right| Rm\kern0.5em \mathrm{is}\kern0.5em \mathrm{a}\kern0.5em \mathrm{small}\kern0.5em \mathrm{module}\right\}$ is the Rad-small submodule. We study basic properties of these modules regarded as a proper generalization of semilocal modules. In particular, we show that the class of ${\mathcal{Z}}^{\ast }$ -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring R is ${\mathcal{Z}}^{\ast }$ -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class $\mathrm{\mathcal{R}}\mathcal{S}$ of all short exact sequences $\mathbbm{E}:0\to M\overset{\psi }{\to }N\overset{\phi }{\to }K\to 0$ such that Im(ψ) has a ${\mathcal{Z}}^{\ast }$ -semilocal in N is a proper class over left hereditary rings. We also study some homological objects of the proper class $\mathrm{\mathcal{R}}\mathcal{S}$.