Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are:
How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of a left and a right ideal, but it seems somehow unsatisfactory...
Is $N$ such that $M \otimes_R N \neq 0$ uniquely determined by $M$, up to isomorphism? If not, can we classify such $N$'s in a reasonable way?
Generally, I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}_R (M,N)$ replaced by $M \otimes_R N$...