Injective module

Question

Let $R=M_n(D)$ the Matrix ring over a division ring and consider $R$ as a left module over itself. Is $R$ an injective module?

I know that $R$ is free, hence it is projective. Is $R$ injective? I were thinking about using Baer's criterion, but don't get it to work. Any hints on that?

Answer 1

Suppose $R$ is not injective as a left module over itself. Then the injective envelope is a proper essential extension: $R\subset E(R)$.

But $R$ is semisimple, so all $R$ modules, including $E(R)$ are semisimple, so $R$ is a direct summand of $E(R)$. But if $R\oplus C=E(R)$, $R$ isn't an essential submodule unless $C=\{0\}$, a contradiction since $E(R)$ doesn't properly extend $R$ in that case.

Answer 2

The ring $M_n(D)$ is Morita equivalent to $D$. Over a division ring, every module is free, so is projective. All all modules are projective, then all modules are injective. Therefore over $M_n(D)$, every module is injective.