Let $(A,+)$ be a nontrivial abelian group (with the neutral element "$0$") and $\mathrm{End}(A)$ the ring of endomorphisms $\phi : A \to A$, defined with the "pointwise function addition", $$(f+g)(a) \equiv f(a)+g(a)$$ and the function composition $f \circ g$. If $A$ is finite, say with $|A| = mn$, where $m$ and $n$ are integers strictly greater then $1$, then we can always easily find zero divisors in $\mathrm{End}(A)$. For example, a pair of functions $f(a) = m\cdot a$ and $g(a) = n\cdot a$ (where I'm using abbreviation $m\cdot a = a + \dots + a$, $m$ times).
But what if $(A,+)$ is a finite group of prime order, or if it is not finite at all: Does the ring $\mathrm{End}(A)$ necessarily have zero divisors?
Note: Here I'm not taking into account the trivial answer, namely a pair of "zero maps" $z(a) \equiv 0$.