Show that an algebra is semisimple if and only if every -module is completely reducible.
I know that if is semisimple then every irreducible -module is isomorphic to a submodule of ( is viewed as a module over itself). Then the submodules of are direct summands (there exists another submodule such that the direct sum equals ).
Is it true that if a module is completely reducible, then all of its submodules are as well?
For the other direction I don't have any good ideas. How about you guys?