# Answer to Question #17183 in Abstract Algebra for Melvin Henriksen

Question #17183

Let k be a field of characteristic zero, and let R be the Weyl algebra A1(k) with generators x, y and relation xy − yx = 1. Let p(y) ∈ k[y] be a fixed polynomial. Show that R → End(Vk) is injective but not an isomorphism.

Expert's answer

The natural map

*α*:*R →*End(*Vk*) must beinjective since*R*is a simple ring. However, since*Vk*is infinite- imensional, End(*Vk*) is not a simple ring. Therefore,*ϕ*is not an isomorphism.
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