I'm working my way through a proof that, given a ring $\langle R, +, \cdot, \theta \rangle$, $\langle \text{End}(R), +, \circ, z \rangle$ is also a ring (where $z(x)=\theta$ for all $x \in R$). Everything seems fine until I try to prove that every element has an additive inverse. For $f \in \text{End}(R)$, I've defined $-f : R \to R$ by the rule $(-f)(x) = -f(x)$ for all $x \in R$. The trouble comes when I try to prove that this function is also an endomorphism, in particular when I try to show that it preserves the product. For $x,y \in R$ I've got $(-f)(x \cdot y) = -f(x \cdot y) = -f(x) \cdot f(y)$ and $(-f)(x) \cdot (-f)(y) = (-f(x)) \cdot (-f(y)) = f(x)f(y)$, so it seems a lot like the product isn't preserved by $-f$. What am I not seeing here?
Edit:
This exercise seems more fishy the more I look at it, so I'll include its text here:
Let $\langle R, +, \cdot, \theta \rangle$ and $\langle R', +', \cdot', \theta' \rangle$ be rings. We denote the set of all morphisms from $R$ to $R'$ by $\text{Mor}(R, R')$. Is $\text{Mor}(R, R')$ closed under the functional addition: $(f + g)(x) = f(x) + g(x)$? We call a morphism of a ring R into R, itself, an endomorphism and denote $\text{Mor}(R, R)$ with $\mathcal{E}(R)$. Show that $\langle \mathcal{E}(R), +, \circ, z \rangle$ is a ring. (L. E. Sigler Algebra Chapter 2 "Rings: Basic Theory" Section 6 "Morphisms" Exercise 11)
I'd also like to point out that, when listing the ring axioms, this textbook forgets to mention the existence of additive inverses (despite stating that requirement later on).