Let $G$ be a finite group. A theorem of Rim (Proposition 4.9 here) states that a $\mathbb{Z}G$-module $M$ is projective if and only if $M$ is $\mathbb{Z}P$-projective for all Sylow subgroups $P$ of $G$.
Which (or what kind of) rings can we substitute for $\mathbb{Z}$ above such that the statement remains valid?