Duo Property Applied to Powers and Regular Elements

The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called right exp-DR.We investigate the structures of group rings,right quotient rings,matrix rings and(skew)polynomial rings,through the study of right exp-DR rings.In addition,we provide a method of constructing finite non-abelian p-groups for any prime p.

Notes on Noncommutative VNL-rings and GVNL-rings

In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong （Glasgow Math. J., 48（1）（2006）） negatively. Moreover, some new results about VNL-rings and GVNL-ringsare also given.

BOUNDARY ELEMENT ANALYSIS OF CONTACT PROBLEMS USING ARTIFICIAL BOUNDARY NODE APPROACH

An improved version of the regular boundary element method, the artificial boundary node approach, is derived. A simple contact algorithm is designed and implemented into the direct boundary element, regular boundary element and artificial boundary node approaches. The exisiting and derived approaches are tested using some case studies. The results of the artificial boundary node approach are compared with those of the existing boundary element program, the regular element approach, ANSYS and analytical solution whenever possible. The results show the effectiveness of the artificial boundary node approach for a wider range of boundary offsets.

ZEROS OF BRAUER CHARACTERS

The authors obtain a sufficient condition to determine whether an element is a vanishing regular element of some Brauer character. More precisely, let G be a finite group and p be a fixed prime, and H = G′O^P′（G）; if g ∈ G^0 - H^0 with o（gH） coprime to the number of irreducible p-Brauer characters of G, then there always exists a nonlinear irreducible p-Brauer character which vanishes on 9. The authors also show in this note that the sums of certain irreducible p-Brauer characters take the value zero on every element of G^0 - H^0.

Some remarks on one-sided regularity

A ring is said to be right （resp., left） regular-duo if every right （resp., left） regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of （one-sided） regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.

On the Ranks of Semigroups of Transformations on a Finite Set with Restricted Range

Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ（X, Y） = {α∈ ΡΥ（X） ： Xα Y}, Υ（X, Y） = ΡΥ（X, Y） ∩Υ（X） and Ι（X, Y） = ΡΥ（X, Y） ∩ Ι（X）. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green＇s relations of Υ（X,Y）. In this paper, we present analogous results for bothΡ Υ（X, Y） and Ι（X, Y）. For a finite set X with ｜x｜≥ 3, the ranks of ΡΥ（X） = ΡΥ（X, X）, Υ（X） = Υ（X, X） and.Ι（X） = Ι（X, X） are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ（X,Y）, Υ（X, Y） and Ι（X, Y） for any proper non-empty subset Y of X.

Green＇s Relations on Semigroups of Transformations Preserving Two Equivalence Relations

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, letTF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.