Answer to Question #23568 in Abstract Algebra for jeremy

Question

Show that for any finite-dimensional k-algebra R and any field extension K ⊇ k, (rad R)K ⊆ rad(RK).

Answer 1

It suffices to show that (rad R)^Kannihilates any simple left R^K-module V . But, Vis a composition factor of M^K for some simple left R-moduleM. Since (rad R)M = 0, it follows that (rad R)^KM^K= 0. But then (rad R)^KV = 0, as desired.