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 .
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.