Some Results on v-Multiplication Rings

Abstract

A family Ω of valuations of the field K is said to be of finite character if only a finite number of valuations are non-zero at any non-zero element of K. If w ∈ Ω has ring and maximal ideal , then A = ∩w∈Ω is said to be defined by Ω and ∩ A is a prime ideal called the centre of w on A and denoted by Z(w). If = Az(w), then w is said to be an essential valuation for A. A domain defined by a family of finite character in which every valuation is essential is called a ring of Krull type.

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

“…[15, page 427], [17], [29]) and so, in particular, of Prüfer domain. When ⋆ = d (where d is the identity (semi)star operation on D) the PdMDs are just the Prüfer domains.…”

Essential domains and two conjectures in dimension theory

“…[31, Example 2.1]). Finally, recall that a Krull-type domain is a PVMD in which no nonzero element belongs to an infinite number of maximal t-ideals [21]. We have thus the implications shown in the diagram within the family of Krull-like domains:…”

“…A finite-dimensional domain R is said to be Jaffard if dim(R[X 1 , ..., X n ]) = n + dim(R) for all n ≥ 1; equivalently, if dim(R) = dim v (R) [1,4,14,19,27]. The class of Jaffard domains contains most of the well-known classes of finitedimensional rings involved in dimension theory of commutative rings, such as Noetherian domains [29], Prüfer domains [19], universally catenarian domains [3], and stably strong S-domains [28,30] [21,26,31,34]; as a matter of fact, these mainly arise as polynomial rings over Prüfer domains or as pullbacks, and both settings either yield Jaffard domains or turn out to be inconclusive (in terms of allowing the construction of counterexamples) [1,15]. In order to find the missing link, one has then to dig beyond the context of PVMDs.…”

Ring-Theoretic Properties of PvMDs

“…A Prüfer v-multiplication domain, for short a PvMD, is a domain whose localizations at tmaximal ideals are valuation domains [22]. For this reason, the ideal-theoretic properties of valuation domains globalize to t-ideals of PvMDs and several properties of ideals of Prüfer domains hold for t-ideals of PvMDs: for example a domain is a PvMD if and only if each t-finite t-ideal is t-invertible.…”