Let $R$ be a ring and $M$ an $R$-module. Let $E$ be the set of endomorphisms of $M$, that is $R$-module homomorphisms $M → M$. Endow $E$ with the multiplication operation given by composition of morphisms, and with the addition operation given by $(f + g)(m) = f(m) + g(m)$.
(a) Prove that $E$ is a (noncommutative, in general) ring.
(b) Determine the ring of endomorphisms $E$ for the $R$-module $\frac{R}{I}$, where $I$ is an ideal of $R$.
I think I was able to do part (a) but I'm stuck with (b) and unfortunately don't have any working to show you guys. I would really appreciate it if someone could show me how to do part (b).
I am going to assume that "ring" means a "commutative ring with one". I am going to denote the residue class $m + I = [m]$.
Let $f: R/I \rightarrow R/I$ be an $R$-module homomorphism. Then $f$ is completely specified by its value on $[1]$, for $f( [m] ) = m \cdot f([1])$. Furthermore, $(f+g)( [1] ) = f([1]) + g([1])$ and $f(g([1])) = f([1]) g([1])$.
In summary, we have that $\mathrm{End}_R(R/I) = R/I$.