Module homomorphisms

Question

Let $A$ be a $k$-algebra ($k$ is a field), and let $M$ be an (right) $A$-module.

Let $L\leq M$, is it true that $\exists \phi\in\operatorname{Hom}_k(M,k)$ s.t $\phi(x)=0\iff x\in L$?

If not, why (here in the answer) there is a bijection between the submodules of the dual module and th submodules of the original module?

Thanks in advance.

Answer 1

Not sure why you think this should be true. Let $k$ be a finite field, $A=k[x]$ its polynomial ring, which is obviously an algebra over $k$. Then for $M=A$ and $L=\{0\}$ there can't be any such homomorphisms, because there are no injective maps $M\to k$.