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 ?