Are there integral domains of dimension two with exactly four primes? How do they look like?

Question

I’m trying to think of an example for an integral domain of dimension two with exactly four prime ideals – I fail to find one. Does such a ring exist?

Furthermore, is there a local ring with this property? Is any such ring local?

Answer 1

I don't know an example off the top of my head, but there is a quite general existence theorem. Namely, if $P$ is any finite poset, then there exists a commutative ring $R$ whose poset of prime ideals is isomorphic to $P$. So in your case, you could take $P$ to be $\{a,b,c,d\}$ with $a<b<c$ and $a<d$ (for instance). You then get a two dimensional ring $R$ with exactly four primes and one minimal prime, and modding out the minimal prime gives a domain. This example is not local, since it has two different maximal ideals ($c$ and $d$), but you could make it local by changing the poset so that $d<c$.

This existence theorem is a special case of an even more general theorem of Hochster characterizing the topological spaces that can be Spec of a ring. See Ring with spectrum homeomorphic to a given topological space.