Matrix Ring over Semisimple Ring

Question

Let $R$ be a semisimple ring, and let $M_n(R)$ denote the ring of $n \times n$ matrices over $R$. I am trying to show that $M_n(R)$ is semisimple.

So far, I have that $M_n(R) = \bigoplus_i L_i$, where $L_i$ is the ideal of matrices with all $0$ entries outside the $i$-th column. Since the $L_i$ are minimal, we have that $M_n(R)$ is semisimple.

But this doesn't seem right, since I haven't used the assumption that $R$ is semisimple - this "proof" would work for any ring $R$. Where is the mistake in my proof?

Answer 1

$L_i$ are isomorphic to free modules over $R$ which can be written as the direct sum of simple submodules since $R$ is semisimple. You need the assumption that $R$ is semisimple since $L_i$ are not simple modules.

Answer 2

Use the following two simple observations together with Wedderburn's theorem.

• A ring of matrices over the ring of matrices over a ring R is simply a ring of (larger) matrices over R.

• A ring of matrices over a direct product of rings is just the direct product of the matrix rings of the factors.