Answer to Question #17177 in Abstract Algebra for Melvin Henriksen
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.
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 dimk V <∞,this would imply that dimkR < ∞. Therefore, V must be infinite-dimensional over k.
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