Answer to Question #17103 in Algebra for john.george.milnor

Question

Let R be a semisimple ring. Show that any quotient ring of R is semisimple.

Answer 1

Write R = R₁ ×· · ·×R_r where the Ri’s are the simple components of R. Consider any quotient ring R/I of R. After a reindexing, we may assume that I = R₁ ⊕· · ·⊕R_s for some s ≤ r. Therefore, R/I ∼ R_s+1 ×· · ·×R_r is a semisimple ring.