Unique factorization property of non-unique factorization domains

摘要

Let D be an integral domain, D[X] be the polynomial ring over D, t be the so-called t-operation on D, and t-Spec(D) be the set of prime t-ideals of D. A nonzero nonunit of D is said to be homogeneous if it is contained in a unique maximal t-ideal of D. We say that D is a homogeneous factorization domain (HoFD) if each nonzero nonunit of D can be written as a finite product of pairwise t-comaximal homogeneous elements. In this paper, among other things, we show that (1) a Prufer v-multiplication domain (PvMD) D is an HoFD if and only if D[X] is an HoFD (2) if D is integrally closed, then D is a PvMD if and only if t-Spec(D[X]) is treed, and (3) D is a weakly Matlis GCD-domain if and only if D[X] is an HoFD with t-Spec(D[X]) treed. We also study the HoFD property of A + XB[X] constructions, pullbacks, and semigroup rings.