Tensor products of simple modules over algebras Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. 
We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and $BM=0$. Let $M\otimes_{\mathbb{C}} N$ be the tensor product of $A\oplus B$-modules $A$ and $B$.
${\bf My\ Question:}$ Is $M\otimes_{\mathbb{C}} N$ still a simple $A\oplus B$-module? What conditions on $A$ and $B$ will make this result true? Thanks very much in advance!
 A: The following counterexample is available over a field $k$ other than $\mathbb{C}$. Let $A = B = M = N = L$ be a field extension of $k$. Then you want to ask (after the correction in the comments) whether $L \otimes_k L$ is a simple module over itself. For a commutative ring this is equivalent to being a field, but $L \otimes_k L$ is almost never a field. For example, if $L$ is a finite Galois extension, then $L \otimes_k L$ is a product of $\deg L$ copies of $L$. 
When $k = \mathbb{C}$ we can attempt to use $L = \mathbb{C}(t)$ but I don't know off the top of my head what $\mathbb{C}(t) \otimes_{\mathbb{C}} \mathbb{C}(t)$ looks like. Probably it's not a field either. 
On the other hand, this is true if $A$ and $B$ are both finite products of matrix algebras over $\mathbb{C}$; for example, it's true if $A$ and $B$ are both group algebras over $\mathbb{C}$ of finite groups. 
A: If $M$ and $N$ are finite dimensional, the tensor product $M \otimes_{\mathbb{C}} N$ is a simple module over $A \otimes_{\mathbb{C}} B$, and all simple finite dimensional representations of the latter are of this kind. This is Thm. 3.10.2 (pp. 56--57) of the book by P. Etingof et al (http://www.amazon.de/Introduction-Representation-Student-Mathematical-Library/dp/0821853511). Of course, the drafts of the manuscript have different numberings so the reference only holds for the published version. This was also posted as an intended answer for  Tensor product of two simple modules, where I forgot to mention the algebraic closedness hypothesis on the field $k$. 
