∑–pure—injective modules over a commutative prüfer ring

We give the complete classification of ∑-pure-injective modules over a commutative Prüfer ring.     

Prüfer conditions in the Nagata ring and the Serre’s conjecture ring

The Nagata ring R(X) and the Serre’s conjecture ring R?X? are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R?X?. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38 Glaz, S., Schwarz, R. (2011). Prüfer conditions in commutative rings. Arab. J. Sci. Eng. (Springer) 36:967–983.[Crossref], [Web of Science ®] [Google Scholar]] related to the transfer of Prüfer conditions to these two constructions.     

Vaught’s conjecture for some meager groups

AssumeG is a superstable group ofM-rank 1 and the division ring of pseudo-endomorphisms ofG is a prime field. We prove a relative Vaught’s conjecture for Th(G). When additionallyU(G) =ω, this yields Vaught’s conjecture for Th(G). Research supported by KBN grant 2 P03A 006 09.     

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     

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.     

Injective Modules over Prüfer v-Multiplication Domains

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prüfer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prüfer v-multiplication domain satisfying an almost maximality-type property.     

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.     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.     

Dedekind’s linear independence theorem over commutative semirings

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields.     

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.     

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).     

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.     

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.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

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...     

László Fuchs’s contributions to commutative ring theory

In commemoration of the 90th birthday of Professor Laszlo Fuchs, this article gives a short account of some of his contributions to commutative ring theory.