Cokernel of a module homomorphism

Question

Let $A$ a $K$-algebra. Let $M$, $N$ $A$-modules and $f:M\rightarrow N$ a module homomorphism. The cokernel of $f$ is $Cokerf=N/Imf$ I define a homomorphism $\rho:N\rightarrow N/Imf$ by $\rho(n)=n+Imf$.

Let $Z$ an $A$-module and $g:N\rightarrow Z$ a module homomorphism such that $g\circ f=0$. I want to prove that exist a module homomorphism $g':N/Imf\rightarrow Z$ such that $g=g'\circ\rho$. How I can define $g$ and $g'$?

Answer 1

I suppose you mean ‘how to define $g'$ in function of $g$’, since $g$ is given.

Well, as usual, given a coset $n+\operatorname{Im}f$, with representative $n$, you set $$g'(n+\operatorname{Im}f)\stackrel{\text{def}}{=} g(n)$$ after you've checked this definition does not depend on the representative $n$.

Indeed, if $n'=n+f(m)$ for some $m\in M$, then $$g(n')=g(n)+g(f(m))=g(n)$$ since $g\circ f=0$.