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

Article

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 $$\mathrm{\mathcal{R}}\mathcal{S}$$.

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