Answer to Question #18199 in Abstract Algebra for Melvin Henriksen

Question

Let R be a k-algebra where k is a field, and M,N be left Rmodules, with dimkM <∞. It is known that, for any field extension K ⊇ k, the natural map θ : (HomR(M,N))K −→ HomRK(MK,NK) is an isomorphism of K-vector spaces. Replacing the hypothesis dimkM < ∞ by “M is a finitely presented R-module,” give a basis-free proof for the fact that θ is a K-isomorphism.