Unique factorization property of non-unique factorization domains

Abstract

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