# About dual of finitely generated projective module

Let $$M$$ be a finitely generated projective module, $$x \in M$$ and $$x \neq 0$$. Then is it true that there is $$g \in M^*$$ such that $$gx \neq 0$$?

If yes how to prove it? For vector space dual this result is true, but what about projective module?

Also if $$f \in M^*$$, $$f \neq 0$$ then $$fy \neq 0$$ is also true or not ?

The answer to your first question is yes, since finitely generated projective modules are reflexive (see here), meaning that the canonical injective map $$M\to M^{**}$$ is an isomorphism. Hence if $$x\neq 0$$, then $$x\neq 0\in M^{**}$$, so there is $$g\in M^*$$ such that $$gx=\langle g, x\rangle\neq 0$$.
I don't know what is $$y$$ in your second question. If you means there exists such an $$y$$, then it's true by definition of $$f\neq 0$$.