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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 $M \otimes_R N$...

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    The tensor product is functorial, so $N$ such that $M \otimes_R N$ is necessarily determined by $M$ up to isomorphism. Are you asking how to effectively list such $N$?2012-08-22
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    @QiaochuYuan: Well, yes. And first of all, I am asking whether such $N$ is unique, as an analogy with Schur's lemma would suggest. Generally, I think that due to lack of some kind of reflexivity, there may be different modules admitting a nondegenerate bilinear pairing with $M$, but I don't know what happens if we want just simple ones...2012-08-22
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    Ah, sorry, I misinterpreted the meaning of "unique."2012-08-22

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