Artinian rings with Morita duality

Two theorems are established which properly extend the ckss of artinian rings with Morita duality. A question of Anh about QF-rings is answered in the negative. Project supported by the National Natural Science Foundation of China (Grant No. 19771016).     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

The structure of zero-divisor semigroups with graph K n ○K 2

This paper determines all commutative zero divisor semigroups whose zero divisor graph is a complete graph (finite or infinite), or a complete graph (finite or infinite) with one additional end vertex, and gives formulas for the numbers of all such semigroups with n elements. The research of T. Wu is supported by the National Natural Science Foundation of China (Grant No. 10671122) and the Natural Science Foundation of Shanghai (Grant No. 06ZR14049).     

Morita Context and Morita—Like Equivalence for the xst—Rings

The notion of the xst-rings was introduced by Garcfa and Marfn [5] in 1999.In this paper,we cousider Morita context,Morita-like equivalence and the exchange property for the xst-rings.The results of the first Morita theorem are generalized to the xst-rings.So we obtain an important Morita-like equivalence of the xst-rings,from which,as an immediate consequence,we deduce the main result of Xu-Shum-Turner [4] and the standard Morita equivalence,A-Mn(A),for a unital ring A.Moreover,we describe the properties of those well-known intermediate matrix rings,and show that the exchange property for a unital ring A coincides with the one for any Mn(A) as well as any intermediate matrix ring sitting between FM1(A) and FCг(A),which is an extension of a well-known result obtained by Nicholson[7].     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

Finite p-Groups Whose Abelian Subgroups Have a Trivial Intersection

A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given.     

可分解半群的Morita等价

设S,R是可分解半群.记US-FAct={sM∈S-Act|SM=M且SHoms(S,M)≌M],给出了范畴US-FAct与UR-FAct等价的刻划;S分别强Morita等价于一个夹层半群、局部单位半群、幺半群和群的条件;S是完全单半群当且仅当S强Morita等价于一个群且对任何指标集I,S SHoms(S,i∈I S)→i∈I S,s t·f→(st)f,是同构.     

Morita Invariance and Maximal Left Quotient Rings

Abstract

We prove that under conditions of regularity the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. We show that if R and S are two non-unital Morita equivalent rings then their maximal left quotient rings are not necessarily Morita equivalent. This situation contrasts with the unital case. However we prove that the ideals generated by two Morita equivalent idempotent rings inside their own maximal left quotient rings are Morita equivalent.     

Two Kinds of Numbers and Their Applications

C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.     

On Best Approximations from RS-sets in Complex Banach Spaces

The concept of an RS-set in a complex Banach space is introduced and the problem of best approximation from an RS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established,     

The lattice generated by some subvarieties of completely regular semigroups

Using a construction theorem of cryptogroups and congruence methods, we determine an 18-element lattice, generated by {NOBG, ROBG, OBG, NBA, RBA, BA}, of subvarieties of completely regular semigroups. This work was supported by the National Natural Science Foundation of China (Grant No. 10871161) and the Research Foundations for Doctor Programme of Southwest University (Grant Nos. SWNUB2005020, SWNUB2008001)     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

A Note on Linear Extension of Into-Isometries Between Two Unit Spheres of Atomic AL^p-Space （0 〈 p 〈∞）

In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.     

Modified likelihood ratio test for homogeneity in bivariate normal mixtures with presence of a structural parameter

This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.     

On the Block Independence in G-Inverse and Reflexive Inner Inverse of A Partitioned Matrix

By applying the multiple quotient singular value decomposition QQQQQ-SVD, we study the block independence in g-inverse and reflexive inner inverse of 2× 2 partitioned matrices, and prove a conjecture in [Yiju Wang, SIAM J. Matrix Anal. Appl., 19（2）, 407-415（1998）].     

Finite groups in which elements of the same order outside the center are conjugate

We prove that if G is a finite group in which the elements of the same order outside the center are conjugate,then either G is abelian or G(?)S_3.     

Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures

The boundedness in Lebesgue spaces for commutators generated by multilinear singular integrals and RBMO(μ) functions of Tolsa with non-doubling measures is obtained, provided that‖μ‖=∞and multilinear singular integrals are bounded from L1(μ)×L1(μ)to L1/2,∞(μ).     

On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

LetΔ_1(x;φ) be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments ofΔ_1(x;φ) and derive an asymptotic formula for the 3-rd, 4-th and 5-th power moments by using Ivic's large value arguments and other techniques.     

Rings of the Morita Contexts and Semihereditary Maximal V-Orders

Some necessary and sufficient conditions are given for rings of the Morita contexts to be semihereditary maximal V-orders, Prüfer rings and Dubrovin valuation rings.