# Answer to Question #17177 in Abstract Algebra for Melvin Henriksen

Question #17177

Let R be a simple, infinite-dimensional algebra over a field k. Show that any nonzero left R-module V is also infinite-dimensional over k.

Expert's answer

The left action of

*R*on*V*leads to a*k*-algebra homomorphism*ϕ*:*R →*End(*Vk*).Since*V <>*0 and*R*is a simple ring,*ϕ*must be one-one. If dim*k V <∞*,this would imply that dim*kR < ∞*. Therefore,*V*must be infinite-dimensional over*k*.
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