Extending properties of \(z\)-closed projection invariant submodules

Abstract

In this article, we define a module \(M\) to be \(ZPG\) if and only if for each \(zp\)-submodule \(X\) of \(M\) there exists a direct summand \(D\) such that \(X\cap D\) is essential in both \(X\) and \(D\). We investigate structural properties of \(ZPG\) modules and locate the implications between the other extending properties. Our focus is the behavior of the \(ZPG\) modules with respect to direct sums and direct summands. We obtain the property is closed under right essential overring and rational hull.