Symmetric modules over their endomorphism rings

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=\operatorname{End}_R(M)$. In this paper, we study right $R$-modules $M$ having the property for $f,g \in \operatorname{End}_R(M)$ and for $m\in M$, the condition $fgm = 0$ implies $gfm = 0$. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting.