Suppose $R\to S$ is a ring homomorphism. Let $M$ be an $S$-module. Suppose $M$ is projective as an $R$-module. Is $M$ projective as an $S$-module? If not, what additional hypothesis would suffice (for example, $S$ projective over $R$)?
A couple of thoughts. It is easily seen that $M\otimes_R S$, the extension of scalars of the restriction of scalars of $M$, is $S$-projective. Also, $M$ is a direct summand of $M\otimes_R S$ as an $R$-module.