Why is the ring of integers initial in Ring? In Algebra Chapter 0 Aluffi states that the ring $\Bbb{Z}$ of integers with usual addition and multiplication is initial in the category Ring. That is for a ring $R$ with identity $1_{R}$ there is a unique ring homomorphism $\phi:\Bbb{Z}\rightarrow{R}$ defined by $\phi(n)\mapsto{n\bullet1_{R}}$ $(\forall{n\in\Bbb{Z}})$which makes sense for rings such as $\Bbb{Q},\Bbb{R},\Bbb{C}$ which have $\Bbb{Z}$ as a subring but I fail to see how $\phi$ holds when the codomain is a ring which doesn't contain $\Bbb{Z}$. If someone could provide examples of ring homomorphisms from $\Bbb{Z}$ to rings other than the rings mentioned above I would appreciate it.

Any such morphism $f$ satisfies $f(n)=f(\sum_{i=1}^n 1) = \sum_{i=1}^n f(1) = \sum_{i=1}^n 1_R = n 1_R$. You have the unicity. The existence is trivial, just define it this way, by the previous formula. So you have the definition for any ring $R$ with unit. ;-) Note that the image of $\mathbf{Z}$ is always in $R$'s center.
Remark. $n 1_R$ for $n\in\mathbf{N}$ simply means $0$ if $n=0$, $\sum_{i=1}^n 1_R$ if $n>0$, and the opposite if $n<0$. I precise this as I think that this is the only thing bothering you.
