A variant of the Reynolds operator

Let $G$ be a linearly reductive group over a field $k$, and let $R$ be a $k$-algebra with a rational action of $G$. Given rational $R$-$G$-modules $M$and $N$, we define for the induced $G$-action on Hom$_{R}(M,N)$ a generalized Reynolds operator, which exists even if the action on Hom $_{R}(M, N)$ is not rational. Given an $R$-module homomorphism $M \rightarrow N$, it produces, in a natural way, an $R$-module homomorphism which is $G$-equivariant. We use this generalized Reynolds operator to study properties of rational $R$-$G$ modules. In particular, we prove that if $M$ is invariantly generated (i.e. $M = R \cdot M^{G}$), then $M^{G}$ is a projective (resp. flat) $R^{G}$-module provided that $M$ is a projective (resp. flat) $R$-module. We also give a criterion whether an $R$-projective (or $R$-flat) rational $R$-$G$-module is extended from an $R^{G}$-module.