Compute Hom(F,M) over a commutative ring, where F is free of finite rank

Let R be a commutative ring with 1, let F be the free left unital R-module of finite rank n, and let M be a left unital R-module. Prove that \mathsf{Hom}_R(F,M) \cong_R M^n.


Recall by Theorem 6 in D&F that F \cong R^n, since F has finite rank. Now \mathsf{Hom}_R(F,M) \cong_R \mathsf{Hom}_R(R^n,M) \cong_R \mathsf{Hom}_R(R,M)^n (by this previous exercise) \cong_R M^n (by this previous exercise).

Thus \mathsf{Hom}_R(F,M) \cong_R M^n.

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