I want to prove $\mathfrak{a}(M/N) = (\mathfrak{a}M + N) / N$, where $M$ is an $A$-module and $\mathfrak{a}$ is an ideal of $A$.
There will be many ways, for example, define a map $f:\mathfrak{a}M + N \to \mathfrak{a}(M/N)$ 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.