I’m back after a brief hiatus in which I worked through a set of problems on Lie derivatives, flows, and vector fields. At the end of the day, I just never seemed to muster the strength to look at algebra. Here goes.
We need a lemma first (I know. If I carefully had planned this, it would have been taken care of already). Lemma: If R is considered as a left module over itself, then . The natural map to check is by . It is just routine checking that this works. We are in since multiplication gets reversed: .
Artin-Wedderburn: A ring R is semisimple iff R is isomorphic to a direct product of matrix rings over division rings.
We already did the sufficient direction. So assume R is semisimple. Then where the are direct sums of isomorphic minimal left ideals (decompose into all minimal left ideals , and then group isomorphic ones into the ). By our above lemma . As a consequence of Schur’s Lemma, when .
Thus we now have . But the can be decomposed into the isomorphic minimal left ideals and we get .
But by Schur again is some division ring and hence . So .
Note that and that is also a division ring (now just rename these division rings) to get that where the are division rings.
We immediately get some nice corollaries. One is that a ring is left semisimple iff it is right semisimple, since a ring is left semisimple iff its opposite ring is right semisimple. Another is that a commutative ring is semisimple iff it is a product of fields.
Next time I’ll do the “uniqueness” part and start on some of the group ring and representation theory consequences (of which there are many).