As a proper generalization of injective modules in term of supplements, we say that a module $M$ has \emph{the property} (SE) (respectively, \emph{the property} (SSE)) if, whenever $M\subseteq N$, $M$ has a supplement that is a direct summand of $N$ (respectively, a strong supplement in $N$). We show that a ring $R$ is a left and right artinian serial ring with $\operatorname{Rad}(R)^2=0$ if and only if every left $R$-module has the property (SSE). We prove that a commutative ring $R$ is an artinian serial ring if and only if every left $R$-module has the property~(SE).