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