Good Morning everyone. I'm currently having trouble with the following:
Problem:
Show that, if $e$ is an idempotent element of a ring $R$, then the two functors $\text{Hom}(eR,-):\textbf{Mod-R}\to\textbf{Ab},$ and $-e:\textbf{Mod-R}\to\textbf{Ab},$ where $-e$ is defined on the objects as $N_R\mapsto Ne$, are canonically isomorphic.
Attempt:
Since the text asks to find a canonical isomoprhism, my guess was to define a map $\eta_N:\textbf{Ab}\to\textbf{Ab}$ as follows:
$\eta_N(f)(er)=f(er)e\quad \forall f\in\text{Hom}(eR,N_R)$. But then i have to prove that $\eta_N$ is an isomorphism for any right $R-$module $N_R$ and at this point I'm stuck.
Moreover i've tried to write $f(er)e=f(e)re$ because i thought i am working with right $R-$modules morphism, but even at this point i'm not sure how to go ahead.
Any solution or hint will be appreceiated and rewarded. Regards.