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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 ?

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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 $f\neq 0$.

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