Maximal chains of prime ideals of different lengths in unique factorization domains

Abstract

We show that, given integers n 1 , n 2 , . . . , n k with 2 < n 1 < n 2 < · · · < n k , there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths n 1 , n 2 , . . . , n k which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.

Every Finite Poset is Isomorphic to a Saturated Subset of the Spectrum of a Noetherian UFD

“…Or, more precisely, given a finite partially ordered set X, does there exist a Noetherian unique factorization domain A such that X is isomorphic to a saturated subset of the prime spectrum of A? Partial progress on this problem is made in [1] and [6] where it is shown that certain noncatenary partially ordered sets are isomorphic to a saturated subset of the prime spectrum of a Noetherian unique factorization domain. In this article, we find a definitive answer.…”

Noncatenary Noetherian Unique Factorization Domains

“…The answer to this question was not known until Heitmann constructed a noncatenary UFD in [4] in 1993. Later, more examples of families of noncatenary UFDs were constructed (see, for example, [1] and [8]). Finally, in [2] it was shown, that, similar to Heitmann's result in [3] for Noetherian rings, given a finite partially ordered set X, there exists a Noetherian UFD R such that X can be embedded into the prime spectrum of R in a way that preserves saturated chains.…”

“…More specifically, we ask if there exists a Noetherian UFD whose spectrum contains infinitely many disjoint noncatenary subsets. We use ideas from [8], [4], and [1] to show in Theorem 3.3 that such a Noetherian UFD does indeed exist. A consequence of Theorem 3.3 is that there exist Noetherian UFDs A satisfying the property that for infinitely many height one prime ideals P of A, the ring A/P is not catenary.…”