# Answer to Question #23568 in Abstract Algebra for jeremy

Question #23568

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

Expert's answer

It suffices to show that (rad

*R*)*annihilates any simple left*^{K}*R*-module^{K}*V*. But,*V*is a composition factor of*M*for some simple left^{K}*R*-module*M*. Since (rad*R*)*M*= 0, it follows that (rad*R*)*= 0. But then (rad*^{K}M^{K}*R*)*= 0, as desired.*^{K}VNeed a fast expert's response?

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