When an $\mathscr{S}$-closed submodule is a direct summand

TL;DR

This paper investigates conditions under which a direct sum of CLS-modules is itself CLS, generalizes some known results, and corrects a false claim about CS-modules related to module decompositions.

Contribution

It proves that certain relative ojectivity conditions ensure CLS-ness of direct sums and refutes a previous claim about CS-modules and their properties.

Findings

Direct sums of CLS-modules can be CLS under specific conditions.

Tercan's claim about CS-modules and $Z_2(M)$ is shown to be false.

Generalizes known results on module decompositions.

Abstract

It is well known that a direct sum of CLS-modules is not, in general, a CLS-module. It is proved that if $M=M_1\oplus M_2$, where $M_1$ and $M_2$ are CLS-modules such that $M_1$ and $M_2$ are relatively ojective (or $M_1$ is $M_2$-ejective), then $M$ is a CLS-module and some known results are generalized. Tercan [8] proved that if a module $M=M_{1}\oplus M_{2}$ where $M_{1}$ and $M_{2}$ are CS-modules such that $M_{1}$ is $M_{2}$-injective, then $M$ is a CS-module if and only if $Z_{2}(M)$ is a CS-module. Here we will show that Tercan's claim is not true.