Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. If $R$ is a direct summand of $M^{\oplus n}$ for some $n\ge 1$ , then is $R$ a direct summand of $M$?
I can easily prove this if $R$ is local, but have a feeling that it is not true if $R$ is not local, although I cannot find a counterexample.
Please help.