If $V$ is free, show that $f$ is surjective.

Question

Let $R$ be a commutative ring with $1$. Let $V$ and $W$ be $R$-modules.

a) Exhibit a canonical $R$-linear map $f: V^* \otimes V \to R $

b) If $V$ is free, show that $f$ is surjective.

Now for the first part $f:V^* \otimes V \to R$ would be $f(\beta \otimes v)=\beta(v)$ $\forall \beta \otimes v \in V^* \otimes V$

How can I prove the second part?

Answer 1

This is true if $V=R\coprod V_1$,consider the $R$-homomorphism $\alpha:V\rightarrow R$ which maps $(a,b)$ to $a$.then $f(\alpha\otimes (1,0))=1$.

Hence your question is a special case.