Answer to Question #17102 in Algebra for john.george.milnor

Question

Let R be a semisimple ring. Show that any ideal of R is a sum of simple components of R.

Answer 1

Let R = B1 ⊕· · ·⊕Bn, where the Bi’s are ideals of R. Then any left ideal (resp. ideal) I of R has the form I = I1 ⊕· · ·⊕In where, for each i, Ii is a left ideal (resp. ideal) of the ring Bi. Thus statement follows directly from mentioned fact.