Suppose that we have a commutative ring $R$ with an idempotent $e$, and $M$ an $R$-module such that $Me$ is $Re$-projective. I am interested to know under which conditions this implies that $M$ is $R$-projective.
That would be true if $M(1-e)$ is $R(1-e)$-projective, but as Martin remarked that is not true.
Any remark will help. Thanks.
Take any product $R_1 \times R_2$, a projective $R_1$-module $M_1$ and a non-projective $R_2$-module $M_2$. Then $M_1 \times M_2$ is a counterexample.