Let $R$ be a ring. Determine all $R$-module homomorphisms $\varphi:R\rightarrow R$.
For any $\varphi$, $\ker\varphi$ and im $\varphi$ both have to be submodules of $R$. In this case, that makes them ideals of $R$. So every $\varphi$ is a surjective map from $R$ to an ideal of $R$ so, if $I$ is some ideal of $R$ I'm really looking for every $\varphi:R \twoheadrightarrow I$.
That's about as far as I've managed to get. I'm not even sure what form the answer is supposed to take.
Thanks...