Extensions of weakly supplemented modules

Abstract

It is shown that weakly supplemented modules need not be closed under extension (i.e. if $U$ and $M/U$ are weakly supplemented then $M$ need not be weakly supplemented). We prove that, if $U$ has a weak supplement in $M$ then $M$ is weakly supplemented. For a commutative ring $R$, we prove that $R$ is semilocal if and only if every direct product of simple $R$-modules is weakly supplemented.