Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 139 / 2 / c / 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 ${\mathrm{Mat}}_n(R)$-module. Conversely, if $V$ is simple over the matrix ring, $e_{11}V$ is a simple $R$-module andthrough 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