It turns out there is a nice classification for semisimple rings.

Theorem. Any semisimple ring R is a finite product:where each is a division ring and is the ring of n × n matrices with entries in D. Furthermore, the list of is unique up to permutation, and isomorphism class of

We saw that a semisimple ring *R* is a *finite* direct sum of simple submodules (left ideals):

where the simple modules are pairwise non-isomorphic. Schur’s lemma says that for simple modules *M* and *M’*, is zero if *M* and *M’* are not isomorphic, and is a division ring otherwise.

More generally, for a simple module *M*, we have:

where *D* is a division ring.

[ Note: in general, we have and by the universal properties of direct sum and product. When we have finitely many terms, the direct sum *is* the direct product. ]

Next we have the isomorphism , where

[ We need to take the opposite ring since ]

Piecing all these together, we have:

By the above discussion, each term is either 0 (if *i* ≠ *j*) or isomorphic to , where is a division ring. In conclusion:

for division rings

[ We leave it to the reader to prove that matrix product in corresponds to composition of endomorphisms . ]

To show that the set of is unique up to isomorphism, note:

Proposition. The ring , for division ring D, has a unique simple module up to isomorphism, namely .

**Proof**

Indeed, *R* is a direct sum of column vectors. E.g. for *n* = 2, we have:

It is easy to see that the space of column vectors is simple as an *R*-module. On the other hand, since every simple *R*-module occurs as a simple left ideal of *R*, is the only possible simple *R*-module. ♦

Hence, for a ring , there are exactly *k* simple modules up to isomorphism, each simple module occurs times and its endomorphism ring is isomorphic to . This shows that we can recover from the ring *R* itself, so it must be unique.

The theorem also shows:

Corollary. R is semisimple iff its opposite ring is. Another of saying this is: R is “left semisimple” iff it is “right semisimple”.

Coming up next, the most interesting example of semisimple rings: *group rings*.