Answer to Question #17104 in Algebra for john.george.milnor

Question

Let R be a semisimple ring. Show that a simple artinian ring S is isomorphic to a simple component of R iff there is a surjective ring homomorphism from R onto S.

Answer 1

If S ∼ Ri, we can find a surjective ring homomorphism from R to S by utilizing the ith projection of R = R1 ×· · ·×Rr. Conversely, suppose ϕ : R → S is a surjective ring homomorphism. After a reindexing, we may assume that ker(ϕ) = R₁ ×· · ·×R_r−1. Therefore, S ∼ R/ ker(ϕ) ∼ R_r.