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[Community Question] Linear-algebra: About dual of finitely generated projective module

One of our user asked:

Let say $x \in M$ and $x \neq 0$ then is it true that for finitely generated projective module $M$ there is $g \in M^*$ such that $gx \neq 0$. If yes how to prove it. For vector space dual this result is true 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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