= Solution
For a simple $R$-module $S$, matrix units show that every nonzero vector of $S^n$ generates all coordinates, so $S^n$ is a simple $\operatorname{Mat}_n(R)$-module. Conversely, if $V$ is simple over the matrix ring, $e_{11}V$ is a simple $R$-module and
$$
V\cong(e_{11}V)^n
$$
through the maps induced by $e_{i1}$ and $e_{1i}$. These constructions are inverse on isomorphism classes; this is the basic <Morita equivalence> for a matrix ring.
The <Jacobson radical> is the intersection of annihilators of all simple left modules. On $S^n$, a matrix annihilates every vector exactly when each entry annihilates $S$. Intersecting over all simple $S$ gives
$$
J(\operatorname{Mat}_n(R))=\operatorname{Mat}_n(J(R)).
$$
Back to article page
