Are there any homomorphisms from integers into finite rings other than modulo $n$?

Question

Are there any "homomorphisms" from $Z$ onto finite rings other than $Z/nZ$ ? I think if instead of mapping $k$ to $k$ (mod $p$), you map it to $p - (k$ (mod $p$)$)$ and you get $f(-ab) = f(a)f(b)$. But are there any interesting ones?

Answer 1

The kernel of a homomorphism needs to be an ideal. All proper ideals of $\mathbb{Z}$ are principal, i.e., in one-to-one correspondence to elements $n=0,1,2, \dots$.

If you ask about group homomorphisms, the answer is the same.

Of course, I assume implicitly for $n=0$ that you mean $\mathbb{Z} / 0\;\mathbb{Z} :=\mathbb{Z}$. I seems that you forgot about the map $\mathbb{Z} \rightarrow \mathbb{Z}, n \mapsto n$.