a left semisimple ring is also a right semisimple ring and so we just say semisimple. anyway, to prove that if is a division ring, then is semisimple, let be the set of all elements of all of whose columns except for the -th one are zero. it is obvious that it is also easy to see that each is a minimal left ideal of so each is a simple left module and we're done.

we actually have for all and so it can also be proved that any simple left module is isomorphic to