The question is from a Persian text book:
Let $R$ be a commutative local ring with $1$, and $\mathfrak m$ its maximal ideal. If $M$ is a non-zero module over $R$ prove that $\text{Hom}_R(M,R/\mathfrak m) \neq 0$.
I have managed to prove this when $M$ is finitely generated by Nakayama lemma, we use it to show that $M/\mathfrak mM \neq 0$, but I couldn't progress for the case when $M$ is not finitely generated, I just proved that the problem is equivalent to the statement below: $$M \neq \mathfrak mM.$$ Any help or hint is appreciated.
Edit:
Seems like for the case that $M$ is not finitely generated we have the following counter example:
$$R= \mathbb{R}[[x]], M= \sum_{i \in \mathbb{Z}} a_i x^i$$