While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was the neccesary assumptions, but I failed, since I don't know any "toy"-examples of such rings.


I know only one example of this kind ($\mathbf{Q} [x]/ (x^2) $ ) but it's not a domain.


So,


**Are there any infinite domains with finite number of prime ideals?**


**If no, then are there any infinite domains with finite (but nontrivial) number of prime elements?**


I am interested in noncommutative examples as well.
Sorry if this question is too elementary.



Infinite domain with finite number of prime ideals(elements)
For the commutative case. Take the set of all rational numbers whose denominators are coprime with a fixed integer $n$. Then the only prime ideals are generated by the prime divisors of $n$.


More generally, take any PID and take its ring of fractions wrt all the elements coprime with some fixed element; you get a desired ring.


