$\scr{Z^{ *}}$ - semilocal modules and the proper class $\scr{RS}$ - Türkmen, E. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ *}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ *}}$ -supplement $V$in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ *}} (V )$, where $\scr{Z^{ *}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $Rad$-smallsubmodule. 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 $\scr{Z^{ *}}$ -semilocal modules is closed under submodules, direct sums, and factor modules.Moreover, we prove that a ring $R$ is $\scr{Z^{ *}}$ -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 $E :0 \xrightarrow{ψ} M \xrightarrow{φ} K → 0$ such that $Im(ψ )$ has a$\scr{Z^{ *}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the properclass $\scr{RS}$ .