Unique factorization property of non-unique factorization domains II

Abstract

Let D be an integral domain. A nonzero nonunit a of D is called a valuation element if there is a valuation overring V of D such that aV∩D=aD. We say that D is a valuation factorization domain (VFD) if each nonzero nonunit of D can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD D is Schreier, and hence Clt(D)={0}, (ii) if D is a PvMD, then D is a VFD if and only if D is a weakly Matlis GCD-domain, if and only if D[X], the polynomial ring over D, is a VFD and (iii) a VFD D is a weakly factorial GCD-domain if and only if D is archimedean. We also study a unique factorization property of VFDs.