Let $R$ be a ring, $S \subseteq R$ a subring, and $M$ is a (left) $R$-module. If $M$ is projective as an $S$ module, then does it follow that $M$ is projective as an $R$-module?
I guess yes, since $S$-projectivity of $M$ is equivalent to the existence of an embedding $$ M \hookrightarrow S^{\oplus k} $$ for some $k$. But since $S^{\oplus k}$ clearly embeds into $R^{\oplus k}$, we "must" also have $R$-projectivity. But maybe I have done something very stupid . . .
$S=K$ a field, $R=K[x]$ and $M=K[x]/(x^2)$.