Let $S$ and $R$ be two (not necessarily commutative) $k$-algebras for $k$ a field. If I have a $S$-$R$ bimodule $\_SM\_R$, I can form the functor $\_SM\_R\otimes\_R (-):R\text{Mod} \rightarrow S\text{Mod}$. Similarly, I can form $(-)\otimes\_S {\_SM\_R}$ to get a functor from $\text{Mod}S \rightarrow \text{Mod}R$. If there are left adjoints to these tensor product functors, I want to show that $M$ must be projective and finitely generated over $R$ and $S$ respectively. Does anyone know any sources that might contain such an argument? 



Existence of a left adjoint to tensor product implies projectivity
I believe I have worked out an argument: We want the functor $F:R\text{Mod}\rightarrow S\text{Mod}$ by $\_S M\_R \otimes\_R (-)$. By tensor-hom duality, this functor is always a left adjoint so in particular it is right exact. If this functor is also a right adjoint, then it must be left exact which means that $M$ is flat as an $S$-module. In the case that I'm interested in, $R$ is the algebra $A^e$ and $M$ is $A$ as a $k-A^e$ bimodule where $k$ is some ground field. $A$ being flat over $A^e$ as a right module is equivalent to $A$ being separable as a $k$-algebra. 


