Why an initial ring is a domain?

It is a well known fact that $\mathbb{Z}$, the ring of integers, is a domain. On the other hand, $\mathbb{Z}$ is also the initial object in the category Ring. If one defines $\mathbb{Z}$ as the initial object in Ring, is it possible to prove it is a domain without an explicit construction?

Solution 1:

Let $A$ be the initial object. Let $g$ be the unique homomorphism $A\to\Bbb Z$. Consider the homomorphism $f\colon \Bbb Z\to A$ given by $$n\mapsto \underbrace{1+\cdots +1}_n$$ (Admittedly, the existence of this already somehow shows that $\Bbb Z$ is initial - the fact is simply too trivial). Then $f\circ g$ must be the unique homomorphism $A\to A$, i.e., the identity. In particular, $g$ is injective while mapping zero divisors to zero. It follows that $A$ has no zero divisors.