Orthogonality of Generalized (θ,φ)-Derivations on Ideals

In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized (θ,φ)-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.     

A Note on Rings and Modules Graded by Some Special Groups

In this note we prove that the prime madicals of some graded rings and the scoles of some graded modules are also graded. The results on the relations between the prime radical and the graded prime radical are also presented. Moreover, the relations between a graded ring and its initial subring are studied.     

A Note on Ideals of a Three-Dimensional Associative Algebra

In this paper, we shall apply the theory of nilpotent radical to derive some results about ideals of a there-dimensional associative algebra over a field F and of an associative ring of order p~3 where p is a prime. These results are useful in studying the structure of the algebra and the ring.     

关于Г-半环

Just as Nobusawa has extended the concept of ring to the concept of Γ-ring. we introduce in this paper a new concept of Γ-hemiring based on the concept of abelian additive hemigroup. And then, we extend the foundamental results associated with hemiring which are developed by Bourne and Zassenhaus to some analogous results associated with Γ-hemiring, such as the notions of ideal, zeroid, homomorphism and Jacobsonian radical and semi-radical in the case of Γ-hemiring. Finally, we have given some conditions for a Γ-hemiring being a Γ-ring.     

分次单环的结构

A characterization of gr-simple rings is given by using the notion of componentwise-dense subrings of a full matrix ring over a division ring. As a consequence, any G-graded full matrix ring over a division ring is isomorphic to a dense subring of a full matrix ring with a good G-grading. Some conditions for a grading of a full matrix ring to be isomorphic to a good one are given, which generalize some results in: Dascascu, S., Lon, B., Nastasescu, C. and Montes, J. R., Group gradings on full matrix rings, J. Algebra, 220(1999), 709-728.     

强$J$-Semiclean环

We in this note introduce a new concept, so called strongly J-semiclean ring, that is a generalization of strongly J-clean rings. We first observe the basic properties of strongly J-semiclean rings, constructing typical examples. We next investigate conditions on a local ring R that imply that the upper triangular matrix ring T_n(R) is a strongly J-semiclean ring. Also,the criteria on strong J-semicleanness of 2 × 2 matrices in terms of a quadratic equation are given. As a consequence, several known results relating to strongly J-clean rings are extended to a more general setting.     

LIE IDEALS, MORITA CONTEXT AND GENERALIZED （α β）-DERIVATIONS

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson＇s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized （α β）-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.     

Γ-环和广义Γ-环的强幂零性

In this paper, we study the strongly nilpotency of Γ-rings and generalized Γ-rings.(Ⅰ) With what conditions is a strongly nil subring of Γ-rings strongly nilpotent?(Ⅱ) With what conditions is a nil subring of Γ-rings strongly nilpotent?For(Ⅰ), the condition is one of the following conditions:(1) ascending chain condition on the- strongly nilpotent subrings.(2) Noether's condition.(3) Goldie's condition.(4) ascending chain condition on right annihilators and on left annihilators. For(Ⅱ), the conditions is (2) or (3). In order that the above results may contain the corresponding results of associative rings, a new concept, generalized Γ-ring is introduced, so that any associative ring is naturally explained as a generalized Γ-ring. It is proved that the above results are valid in generalized Γ-rings. Thence the well-known results of associative rings are special cases in generalized Γ-rings.     

Sharp Distortion Theorems for a Subclass of Biholomorphic Mappings Which Have a Parametric Representation in Several Complex Variables

In this paper, the sharp distortion theorems of the Fr′echet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in C~n are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are got as well. Thus, some known results in prior literatures are generalized.     

On π-regularity of General Rings

A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left （right） semicentral if for every x ∈ I,xe=exe （ex=exe）.And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N （I） of nilpotent elements in I is an ideal of I and I /N （I） is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.     

Repeated-root cyclic and negacyclic codes over a finite chain ring

We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring.     

Description of Barely Transitive Groups with Soluble Point Stabilizer

We consider a ring whose left modules of finite length are semisimple and prove some results on such a ring. We also consider when such a ring is a left V-ring.     

关于Baer PP和PS环的一些性质(英)

在此文中,我们对Strong-Armendariz环和Baer PP及PS环Ore-扩张R[x,x~(-1)；α]的一些性质进行了讨论研究,并得到了一些结果．主要证明了R是Baer(PP)环当且仅当R[[x]]是Baer(PP)环及R是α-rigid环时,R是Baer(PP,PS)环当且仅当R[[x]]是Baer(PP,PS)环．     

Designing Optimal Algorithms for Solving Banded Triangular Systems on Rings

Parallel complexity results for designing banded triangular solvers are provided. In particular, several lower bounds based on data layout and communication along a ring are derived based on solving such systems using substitution. Lastly, a near-optimal solver for a ring is discussed and provided.     

Modules Whose Endomorphism Rings are Division Rings

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End _R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.     

Study of formal triangular matrix rings

In this paper we carry out a systematic study of various ring theoretic properties of formal triangular matrix rings. Some definitive results are obtained on these rings concerning properties such as being respectively left Kasch, right mininjective, clean, potent, right PF or a ring of stable rank ≤ n. The concepts of a strong left Kasch ring, a strong right mininjective ring are introduced and it is determined when the triangular matrix rings are respectively strong left Kasch or strong right mininjective. It is also proved that being strong left Kasch or strong right mininjective are Morita invariant properties.     

Zero commutativity of nilpotent elements skewed by ring endomorphisms

The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.