Real Prüfer extensions of commutative rings

Let A⊂R be an extension of commutative rings with 1. We show that A is totally real (i.e. all maximal ideals of A are real) and A⊂R is a Prüfer extension if and only if R is totally real and the holomorphy ring H(R/A) of R over A is A. Received: 2 January 2001 / Revised version: 23 April 2001     

Epimorphic extensions and Prüfer extensions¶ of partially ordered rings

The paper contributes to the investigation of epimorphisms in the category of reduced partially ordered rings (porings). Two main questions are considered: 1) Does the set of isomorphism classes of a given poring have a largest element (an epimorphic hull)? 2) Given an epimorphic extension, or even a Prüfer extension, f:A→B of porings: how closely are A and B related to each other? Received: 17 June 1999     

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...     

Primary elements in Prüfer lattices

In this paper we study primary elements in Prüfer lattices and characterize -lattices in terms of Prüfer lattices. Next we study weak ZPI-lattices and characterize almost principal element lattices and principal element lattices in terms of ZPI-lattices.     

Immediate Extensions of Prüfer Rings

We study into questions that naturally arise when Prüfer rings are viewed from the geometry standpoint. A ring of principal ideals which has infinitely many prime ideals and is such that its field of fractions is non-Hilbertian is constructed. This answers in the negative a question of Lang.     

A Characterization of Prüfer Domains

In this article, it is proved that a domain R is a Prüfer domain if and only if it is coherent, integrally closed and FP-id _R (R) ≤ 1.     

Prüfer’s ideal numbers as Gelfand’s maximal ideals

Polyadic arithmetics is a branch of mathematics related to p-adic theory. The authors suggest two non-classical models for the Prüfer profinite completion Z of the ring Z. Firstly, letA be the algebra of all periodic functions on Z, then Z can be defined as the ring of all non-zero morphisms A → C with convolution ring operations equipped with some natural topology. Secondly, let (E, μ) be the maximal ideal space for the Banach algebra of almost periodic functions on Z with the Gelfand topology μ. One can define rings operations in (E, μ) which turn it into a topological ring isomorphic to Z.     

Extending Rings of Prüfer Type in Central Simple Algebras

Let R be a commutative integral domain with field of fractions F and let Q be a finite-dimensional central simple F-algebra. If R is a Prüfer domain then it is still unknown whether or not R can be extended to a Prüfer order in Q in the sense of Alajbegovi? and Dubrovin (J. Algebra, 135: 165–176, 1990). In this paper we investigate a more general class of rings which we call rings of Prüfer type and we will prove an extension theorem for these rings. Under special assumptions this result also leads to an extension theorem for certain Prüfer domains.     

Ascending Nets of Prüfer V-Multiplication Domains

Abstract

Let D be an integral domain, S ? D a multiplicative set such that aD _S  ∩ D is a principal ideal for each a ∈ D and let D ^(S) = ? _s∈S D[X/s]. It is known that if D is a Prüfer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D ^(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D ^((S)) = ? _s∈S D[[X/s]] of D ^(S). Our approach takes care simultaneously of both cases D ^(S) and D ^((S)).     

Maximally Prüfer Rings

In this paper we consider six Prüfer-like conditions on a commutative ring R, and introduce seventh condition by defining the ring R to be maximally Prüfer if R _M is Prüfer for every maximal ideal M of R, and we show that the class of such rings lie properly between Prüfer rings and locally Prüfer rings. We give a characterization of such rings in terms of the total quotient ring and the core of the regular maximal ideals. We also find a relationship of such rings with strong Prüfer rings.     

On Extensions of Semilocal Prüfer Domains

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prüfer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prüfer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prüfer domain. Moreover, we show that any two extensions of a semilocal Prüfer domain are incomparable (by inclusion) in an algebraic extension of fields.     

Prüfer and Morita Hulls of FCP Extensions

We examine in this paper some properties of Morita and Prüfer hulls of FCP and FIP extensions of rings. Dichotomy phenomena appear in the case of a quasi-local base ring. In the general case, we define relative supports that allow us to introduce the concept of direct factorization of an extension and to characterize these hulls.     

On the Simultaneous Basis Property in Prüfer Domains

In this paper we prove that if D is a Prüfer domain such that given a proper invertible integral ideal A of D there exists a nonempty finite set of finitely generated maximal ideals that contain A, then D has the simultaneous basis property. This result is used to study two old open problems: "Does every Prüfer domain have the PA-property?", and "Is every Bézout domain an elementary divisor domain?". We include also a new different proof of the simultaneous basis property for valuations domains.     

Butler Modules Over Prüfer Domains

If R is an integral domain, let  be the class of torsion free completely decomposable R-modules of finite rank. Denote by  the class of those torsion-free R-modules A such that A is a homomorphic image of some C ? , and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ? . Further, let Q  and Q 𝒫 be the respective closures of  and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q  = Q 𝒫, and  = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ?  has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then  = Q  = Q 𝒫 = 𝒫 if and only if R is almost maximal.     

利用Prüfer变换求解微分方程

针对一类二阶非线性常微分方程,利用Prüfer变换将其约化为特殊的一阶常微分方程组,从而使其求解过程得以简化.实例说明应用Prüfer变换求解一类偏微分方程边值问题的技巧.     

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.