K-theory of Prüfer domains

Archiv der Mathematik - In this article, we show that the homotopy invariance of K-theory holds for rings of weak global dimension at most one. Prüfer domains are examples of such rings. We...     

Prüfer ⋆–multiplication domains and ⋆–coherence

Abstract The purpose of this paper is to deepen the study of the Prüfer ⋆–mul-tiplication domains, where ⋆ is a semistar operation. For this reason, we introduce the ⋆–domains, as a natural extension of the v-domains. We investigate their close relation with the Prüfer ⋆-multiplication domains. In particular, we obtain a characterization of Prüfer ⋆-multiplication domains in terms of ⋆–domains satisfying a variety of coherent-like conditions. We extend to the semistar setting the notion of -domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the –domains, the Prüfer ⋆-multiplication domains coincide with the ⋆-domains. Keywords: Star and semistar operation, Prüfer (⋆-multiplication) domain, -domain, Localizing system, Coherent domain, Divisorial and invertible ideal Mathematics Subject Classification (2000): 13F05, 13G05, 13E99     

Nagata-like theorems for almost Prüfer v-multiplication domains

Let D be an integral domain and X an indeterminate over D . We show that if S is an almost splitting set of an integral domain D , then D is an APVMD if and only if both DS and DN(S) are APVMDs. We also prove that if {Dα}α∈I is a collection of quotient rings of D such that D=∩α∈IDα has finite character (that is, each nonzero d∈D is a unit in almost all Dα) and each of Dα is an APVMD, then D is an APVMD. Using these results, we give several Nagata-like theorems for APVMDs.     

Ideal class (semi)groups and atomicity in Prüfer domains

Czechoslovak Mathematical Journal - We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is...     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

Decidability of the theory of modules over Prüfer domains with dense value groups

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains whose localizations at maximal ideals have dense value groups. For Bézout domains, these conditions are also necessary.     

SFT stability via power series extension over Prüfer domains

A ring D is called an SFT ring if for each ideal I of D, there exist a natural number k and a finitely generated ideal such that a ^k ∈J for each a∈I. We show that the power series ring over an SFT Prüfer domain D is again an SFT ring even if D is infinite-dimensional. From this, it follows that every ideal-adic completion of D is also an SFT ring. We also show that is an n-dimensional regular ring. B. G. Kang was supported by Korea Research Foundation Grant (KRF 2002-041-C00008). M. H. Park was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-003-C00003).     

The D+E[Γ⁎] construction from Prüfer domains and GCD-domains

Let $D?E$ denote an extension of integral domains, Γ be a nonzero torsion-free grading monoid with $\Gamma \cap ?\Gamma =\{0\}$, ${\Gamma }^{?}=\Gamma ?\{0\}$ and $D+E[{\Gamma }^{?}]=\{f\in E[\Gamma ]|f(0)\in D\}$. In this paper, we give a necessary and sufficient criteria for $D+E[{\Gamma }^{?}]$ to be a Prüfer domain or a GCD-domain.