# Module homomorphism is well defined

Let M be R-module, and A,N submodules such that $$A \subset N$$ Let $$f:M/A \to M/N$$ be defined by $$f(mA)=mN$$ for $$m\in M$$.

I'm having trouble seeing why exactly this map is well-defined (It should be). This should be quite trivial... Please help. Thanks.

You are mapping $$m+A$$ to $$m+N$$ and have to show that this is independent of the representing element of the equivalence class $$mA$$. So let $$m=n\in M/A$$. In other words, $$m-n=a\in A$$. You have to show that $$f(m)-f(n)\in N$$. So consider $$f(m)-f(n)=f(m-n)=f(a)=a\in A\subset N.$$ So your map is independent of the representing element and well-defined.