Towards a classification of stable semistar operations on a Prüfer domain

We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain R has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of R.     

On the Möbius function of the locally finite poset associated with a numerical semigroup

Let S be a numerical semigroup and let (?,≤ _S ) be the (locally finite) poset induced by S on the set of integers ? defined by x≤ _S y if and only if y?x∈S for all integers x and y. In this paper, we investigate the Möbius function associated to (?,≤ _S ) when S is an arithmetic semigroup.     

On the normalizer of a Gaschütz system of a finite soluble group

The notion of a Gaschütz system of a finite soluble group was introduced by S. F. Kamornikov in 2008 (this is a set of complements of crowns of pairwise nonisomorphic non-Frattini factors of a chief series of the group). In the present paper, properties of Gaschütz systems are investigated. In particular, we calculate the number of Gaschütz systems in a finite soluble group and prove their conjugacy, obtain a connection between mathfrakN mathfrak{N} -prefrattini subgroups and normalizers of Gaschütz systems, and investigate factorizations of the normalizer of a Gaschütz system.     

Invertibility of ideals in Prüfer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush and Zhang in [9 Knebush, M., Zhang, D. (2002). Manis Valuations and Prüfer Extensions I. Lecture Notes in Mathematics, Vol. 1791. Springer.[Crossref] , [Google Scholar]], we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.     

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)     

∑–pure—injective modules over a commutative prüfer ring

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

Umt-domains and domains with prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.     

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.     

Separability of the subgroups of residually nilpotent groups in the class of finite π-groups

Siberian Mathematical Journal - Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient...     

Presentations of the Schützenberger Product of n Groups

ABSTRACT

In this article, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid B _n (k). We show that B _n (k) splits as a semidirect product of the monoid of unitriangular matrices U _n (k) by the group of diagonal matrices. When the semiring is a field, B _n (k) is actually a group and we recover a well-known result from the theory of groups and Lie algebras. Pursuing the analogy with the group case, we show that U _n (k) is the ordered set product of n(n ? 1)/2 commutative monoids (the root subgroups in the group case). Finally, we give two different presentations of the Schützenberger product of n groups G ₁,…, G _n , given a monoid presentation ?A _i  | R _i ? of each group G _i . We also obtain as a special case presentations for the monoid of all n × n unitriangular Boolean matrices.     

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.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

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.