A Connection Between the Reticulation of a Ring of Quotients and the Localization Lattice of the Reticulation of a Commutative Ring

Through this article, R denotes a commutative ring with identity. The aim of this article is to construct a morphism of lattices between the reticulation of a ring of quotients (of a commutative ring with respect to a Gabriel topology) and the localization lattice of the reticulation of the initial ring. The article is structured as follows: The first section introduces all the necessary notions required to simplify the reading of the article. The second section presents some properties of the topology induced by a Gabriel topology on the reticulation L(R). In the third section the morphism δ, which achieves the intended link, is constructed. The last section studies the defined morphism in some particular cases.     

Irreducibility and Reduction Number of Ideals in a One Dimensional Analytically Irreducible Ring

Let R be a one dimensional analytically irreducible ring, and let I be an integral ideal of R. We study the irreducibility and the reduction number of I in relation with the corresponding semigroup ideal v(I) in v(R), where v(R) is the semigroup of values of R. It turns out that, if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We show also that the reduction number of I in R can assume any positive value less than the multiplicity of R and can be different from the reduction number of v(I).     

Local Rings of Rings of Quotients

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a. Partially supported by the Ministerio de Educación y Ciencia and Fondos Feder, jointly, trough projects MTM2004-03845, MTM2007-61978 and MTM2004-06580-C02-02, MTM2007-60333, by the Junta de Andalucía, FQM-264, FQM336 and FQM02467 and by the Plan de Investigación del Principado de Asturias FICYT-IB05-017.     

Extending Ring Derivations to Right and Symmetric Rings and Modules of Quotients

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential.

We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.     

Systems of Quotients of Lie Triple Systems

In this article, we introduce the notion of system of quotients of Lie triple systems and investigate some properties which can be lifted from a Lie triple system to its systems of quotients. We relate the notion of Lie triple system of Martindale-like quotients with respect to a filter of ideals and the notion of system of quotients, and prove that the system of quotients of a Lie triple system is equivalent to the algebra of quotients of a Lie algebra in some sense, and these allow us to construct the maximal system of quotients for nondegenerate Lie triple systems.     

Augmentation Quotients for Complex Representation Rings of Generalized Quaternion Groups

Denote by Q_m the generalized quaternion group of order 4m. Let R(Q_m) be its complex representation ring, and Δ(Q_m) its augmentation ideal. In this paper, the author gives an explicit Z-basis for the Δ~n(Q_m) and determines the isomorphism class of the n-th augmentation quotientΔ~n(Q_m)/(Δ~(n+1)(Q_m))for each positive integer n.     

一般环之分式环

本文给出一般环R之分式环△^-1R的构造和△^-R的理想的一些性质,推广了现有中外教科书的结论．证明了（1）△^-1R中理想有且仅有形式△^-1I,其中△3R．（2）如果P是R的素理想,且△∩P=Ф,则△^-1P是△^-1R的素理想,且θ^-1（△^-1P）=P。     

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Congruences Involving Bernoulli Numbers and Fermat-Euler Quotients

Let B_m be the mth Bernoulli number in the even suffix notation and let q(a, n)=(a^?(n)−1)/n be the Fermat-Euler quotient, where a, n?2 are relatively prime positive integers and ? is the Euler totient function. The main purpose of this paper is to devise a certain congruence involving the Bernoulli number and Fermat-Euler quotient, which leads to several important arithmetic properties of Bernoulli numbers.     

环上的σ-导子

本文研究了σ－导子的扩张问题，并且在本原环上刻化了s－导子。     

Irreducibility of unitary group representations and reproducing kernels Hilbert spaces

We discuss unitary representations of groups in Hilbert spaces of functions given together with reproducing kernels, and in particular irreducibility. Our main focus is on examples, including spherical representations of orthogonal groups, distance-transitive finite graphs, irreducibility of induced representations, the discrete series of SL(2, ), and some representations of PGL(2, _p).The appendix contains examples involving groups acting on rooted trees.     

解析Toeplitz代数的完全不可约性与循环向量

解析Toeplitz代数的完全不可约性与循环向量@曹广福...     

Center,Centroid, Extended Centroid,and Quotients of Jordan Systems

In this article we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.     

Entropy and Arithmetic Quotients for Simple Automorphism Groups of Geometric Manifolds

Let G be a connected simple Lie group, and suppose G acts on a compact real analytic manifold M. Assume that G preserves a unimodular rigid geometric structure, for example a connection and a volume form. The aim of this paper is to describe the measure theoretic structure of such actions, when -rank(G)2.     

Every Commutative Ring Has a Minimal Ring Extension

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E ₁ and E ₂ are nonisomorphic simple R-modules, then R(+) E ₁ and R(+) E ₂ give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ? R × R is a minimal ring extension of R ? R is a field.     

有限交换群的Burnside环的增广理想的连续商群的结构

本文对有限交换群的Burnside环的增广理想的n次幂与n 1次幂之商进行了讨论,完整地给出了这类连续商群的结构.     

The Total Graphs of Finite Rings

In this paper we extend the study of total graphs τ(R) to noncommutative finite rings R. We prove that τ(R) is connected if and only if R is not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite rings R.     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

三角环上的σ-双导子

本文利用极大右商环证明了一类三角环上的σ-双导子可以表示成极值σ-双导子和内σ-双导子之和.