Constructing a module homomorphism

Question

Let $\phi: M \to M'$ be a homomorphism of left R-modules, and let $N' \subset M'$ be a submodule. Construct a natural module homomorphism:

$\tilde{\phi}: M/\phi^{-1}(N') \to M'/N'$

and show that $\tilde{\phi}$ is injective. Also, show that $\tilde{\phi}$ is an isomorphism if $\phi$ is surjective.

Thanks in advance.

Answer 1

Hint: simply set $\widetilde\phi(m+\phi^{-1}(N'))= \phi(m)+N'$ and check it is well defined, and it is an injective module homorphism.

For the surjectivity if $\phi$ is surjective, it comes from a general fact about mappings of sets.