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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.

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The natural transformation is a collection of maps $\eta_N:\text{Hom}(eR,N)\to Ne$ for each object $N$ in Mod-R, satisfying the naturality condition.

Define $\eta_N(f)$ as the element $f(e)$ of $Ne$, and note that the inverse $\eta_N^{-1}(n)$ is the element $r\mapsto nr$ of $\text{Hom}(eR,N)$. You can check that this is an isomorphism, and technically you should also check that this is natural in $N$.

edit:

To check naturality, for all morphisms $\phi:N\to M$ in Mod-R there are induced morphisms $\phi e:Ne\to\ Me$ (which is just a restriction of $\phi$) and $\text{Hom}(eR,\phi) : \text{Hom}(eR,N) \to \text{Hom}(eR,M) $ (postcomposition by $\phi$). The naturality condition we wish to check is $(\phi e) \circ \eta_N=\eta_M \circ \text{Hom}(eR,\phi)$. Unwrapping this a bit more, we are checking that $\phi(\eta_N(f))=\eta_M(\phi\circ f)$, which with the above $\eta$ becomes just $\phi(f(e))=(\phi\circ f)(e)$.

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    I've added a paragraph about [naturality](http://en.wikipedia.org/wiki/Natural_transformation).2011-11-28