Let $R$ be a ring, and let $V$ denote the $R$-module $R$. Determine all homomorphisms $\varphi:V\rightarrow V$.
For any $\varphi$, ker $\varphi$ and im $\varphi$ both have to be submodules of $V$. In this case, that makes them ideals of $R$. So every $\varphi$ is a surjective map from $V$ to an ideal of $R$. $V$ consists of everything in $R$ 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.