I'm studying Morita theory on "Rings and Categories of Modules", Anderson. I have some problems with a Theorem about equivalent rings.
Let $R$ and $S$ be equivalent rings via inverse equivalences $F: {}_{R}Mod \to {}_{S}Mod$ and $G: {}_{S}Mod \to {}_{R}Mod$. Set $P=F(R)$, $Q=G(S)$. Then $P$ and $Q$ are naturally bimodules $_{S}P_{R}$ and $_{R}Q_{S}$.
The theorem states other results but I can't demonstrate this first assertion. In the proof he uses two isomorphisms $R \cong End(_{R}R)$ via right multiplication $\lambda$ for a scalar in $R$ and $End(_{R}R) \cong End(_{S}P)$ via $F$. So there is a ring isomorphism $r \mapsto F(\lambda(r))$ and I have to use this one to define a right multiplication of an element of $P$ for a scalar in $R$ to prove that $P$ is a right $R$-module. He also says that we use the fact that $_{R}R_{R}$ is a bimodule and that $F$ is an additive functor.
I have thought of: $P\times R \to P$ sending $(p,r) \mapsto pF(\lambda(r))$ but i'm not sure of this definition and if the result is in P. How can I prove that P is a right R-module? If someone has some suggestions I'd really appreciate!