Ring structure on $\mathbb{Z}$ Is there a possibilty to define a (nontrivial) ring structure on $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) other than the usual so that $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) with that structure becomes an integral domain (or even UFD, PID or euclidean) but not a field?

Yes, of course.
Let $E$ denote your favourite countable Euclidean domain that is not a field, say $\mathbb{Z}[i].$ Since $E$ and $\mathbb{Z}$ are both countable, there is a bijection $f : \mathbb{Z} \rightarrow E$. Now define a binary operation $+_*$ on $\mathbb{Z}$ by writing
$$\forall x,y \in \mathbb{Z}(x+_*y = f^{-1}(f(x)+_E f(y)))$$
Similarly for all the other ring operations / constants. You'll end up with an ring structure on $\mathbb{Z}$ that is isomorphic to $E$.
