Hereditary properties of direct summands of algebras

We consider subrings S of rings R such that R = S oplus V with V either two sided invariant under multiplication by S or invariant under the commutator with S. We show that some important properties of R are inherited by S under such conditions. One is the FCR-property which says that every finite dimensional representation is completely reducible. Another application gives a characterization (in characteristic zero) of reductive subgroups of reductive groups.