I want to prove $\alpha (M/N) = (\alpha M + N) / N$, where $M$ is an $A$-module and $\alpha$ is an ideal of $A$.
There will be many ways, for example, define a map $f:\alpha M + N \to \alpha (M/N)$ and show that $f$ is an $A$-homomorphism and $ker(f)=N$.
But what is the best simple way to prove it? I don't want to define a map and prove it a homomorphism. It looks similar to 2nd ismomorphism but it is a little different.