chapter  20
12 Pages

UMV-Domains

ByEvan Houston, Muhammad Zafrullah

Let R be an integral domain with quotient field K. If f ∈ K[X ] is an irreducible polynomial, then we call the prime ideal P = fK[X ]∩R[X ] an upper to zero. Uppers to zero have been used by many authors to characterize ring-theoretic properties. For example, it follows from [6, Theorem 19.15] that a domain R is a Pru¨fer domain if and only if R is integrally closed and P MR[X ] for each upper to zero P in R[X ] and each maximal ideal M of R. A corresponding result exists for Pru¨fer v-multiplication domains (PVMDs). A domain R is a PVMD if and only if R is integrally closed and each upper to zero in R[X ] is a maximal t-ideal of R[X ] [10, Proposition 2.6]. This led the authors of the present chapter to isolate this condition on uppers to zero by defining UMT-domains to be those domains in which each upper to zero is a maximal t-ideal [12]. UMT-domains were further studied in [3] and [4]. The purpose of this chapter is to study UMV-domains, domains with the property that each upper to zero is a maximal v-ideal, that is, a maximal divisorial ideal.