I know that if $\displaystyle A$ is semisimple then every irreducible $\displaystyle A$-module is isomorphic to a submodule of $\displaystyle A^{*}$ ($\displaystyle A^{*}$ is $\displaystyle A$ viewed as a module over itself). Then the submodules of $\displaystyle A$ are direct summands (there exists another submodule such that the direct sum equals $\displaystyle A$).

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?