Problem with first isomorphism theorem for modules

Question

I am having a conceptual problem with the first isomomorphism theorem, probaly the question is silly, but I don't know how to precede.

Supose that we have an homomorphism of Modules $f : M \to M$, then by the isomorphism theorem, $M/kerf$ is isomorphic to $f(M)$ so, if $f$ is injective, $M$ is isomorphic to $f(M)$ and then $f$ is surjective. But this is not true for every module, (in fact the question was motivated by the problem to show that this is tru for artinian modules), for example, if we consider $\mathbb{Z}$ as a $\mathbb{Z}$-module, $f:$ $\mathbb{Z} \to $ $\mathbb{Z}$ such that $f(a) = 2a$ is monomorphism but not surjective.

I don't see what I am missing here. I'd appreciate any help.

Thank you

Answer 1

$f(M)\neq M$ necessarily (though there is an isomorphism), so $f$ is not surjective necessarily.

Answer 2

The problem is, that you have only $M \cong f(M)$ and you want to deduce $M = f(M)$ (as subsets of $M$). This is not valid.