If $R$ is a left semisimple ring, then $$R \cong \mathbb{M}_{n_1}(D_1) \times \cdot\cdot\cdot \times \mathbb{M}_{n_r}(D_r)$$ where each $D_i$ is a division ring and $\mathbb{M}_{n_i}(D_i)$ is the matrix ring over $D_i$ , $i = 1, 2, \ldots, r$ . The positive integer $r$ is unique, and so are the division rings (up to permutation).

Some immediate consequences of this theorem:

• A simple Artinian ring is isomorphic to a matrix ring over a division ring.
• A commutative semisimple ring is a finite direct product of fields.

This theorem is a special case of the more general structure theorem on semiprimitive rings.