KG_2 over Rings with Division Rings of Quotients

A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.

KG2 over Rings with Division Rings of Quotients

A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd of integers of K is not a unique-factorization domain. For d 〈 0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = -1,-2,-3,-7,-11,-19,-43,-67,-163. Let Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.

Generalized Derivations in Prime Rings

Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.

(Right)Partial Generalized Automorphisms of Semiprime Rings

In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.

Extensions of Symmetric Rings

We first consider properties and basic extensions of symmetric rings. We next argue about the symmetry of some kinds of polynomial rings, and show that if R is a reduced ring then R[x]/（x^n） is a symmetric ring, where （x^n） is the ideal generated by x^n and n is a positive integer. Consequently, we prove that for a right Ore ring R with Q its classical right quotient ring, R is symmetric if and only if Q is symmetric.

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.

Skew Derivations with Power Central Values on Commutators

Let R be a prime ring with center Z, 5 ： R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if （i） [（δ（[x, y]）, [x, y]]n = 0 for all x, y ∈ R or （ii） [（δ（x）, x]n e Z for all x ∈ R, except some specific cases.

Injective Homogeneity and Homological Homogeneity of the Ore Extensions

In this paper we prove that under some natural conditions, the Ore extensions and skew Laurent polynomial rings are injectively homogeneous or homologically homogeneous if so are their coefficient rings. Specifically, we prove that if R is a commutative Noetherian ring of positive characteristic, then An（R）, the nth Weyl algebra over R, is injectively homogeneous （resp. homologically homogeneous） if R has finite injective dimension （resp. global dimension）.

Duo Property on the Monoid of Regular Elements

We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR.We prove that thepolynomial ring over a ring R is right DR if and only if R is commutative.It is also proved that for a prime number p,the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo,and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.

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.