I am trying to solve the following problem:
Let $k$ be a field, and let $R=M_{n}(k)$, the non-commutative ring of $n \times n$ matrices over $k$.
(a) Give examples of a simple left $R$-module $M$ and of a simple right $R$-module N.
(b) Can you find another simple left $R$-module $M^{\prime}$ which is not isomorphic to $M ?$ (Either find one or explain why there is none such.)
(c) Compute $\operatorname{dim}_{k}\left(N \otimes_{R} M\right)$.
For (a), I know that any simple $R$-module is isomorphic to $k^n$ with the standard action of $R$ on $k^n$ and this gives (b). A reference can be found here. A down-to-earth reasoning on why this is true is to note that any simple $R$-module is at least cyclic so that $M=Rv$ for some $v\in M$. Then I can identify $v$ with a vector in $k^n$ and the action of $R$ on $v$ as the action of a matrix on the vector in $k^n$ via left multiplication. I think this makes some sense at least...
Going back to (a), the idea I have is to specify the action on the left and right on some vector of $k^n$. So for instance, I take $M=Re_1$ where $e_1$ is the standard basis vector in $k^n$.
However, I am stuck on how to approach (c). So the tldr of what I would like is this:
- Can I have some hints on (c)?
- Does there exist a straightforward proof that the only simple modules are $k^n$ that does not use Morita equivalence?
- Does my "idea" for giving a simple left $R$-module $M$ and a simple right $R$-module $N$ work?
