理想上的广义$(\theta,\phi)$-导子的正交性

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.     

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.     

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.     

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.     

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.     

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.     

Lie Homomorphism on the Associative Ring

Is it true tliat Lie homomorpHisin (isomorphism) \phi of a ring R into a ring R' is a homomorphism (isomorpliism) ? Herstein,I. N. and Kleinfeld, E. and Martindal, W. S.. obtained some results for simple ring and primitive ring,. In this paper I shall study the Lie homomorphism on the associative ring and arrive at the following main conclusion: Theorem. Suppose that R,R' are both associative rings with 1,and the center ofR’ does not contain zero divisor, where R’ is not of oharaoteristio 2 or 3. If \phi is a 3-Lie homomorphism of R onto R',then \phi must be either a homoinorpiiisin or the negative of an anti-homomorphism of R onto R'. Theorem. Suppose that R is an associatiye ring and $R'\ne {0}$ is a prime ring,where (R', +) does not contain elements of the period 2 or 3. If \phi is a 3-Lie homomorphism of R onto R',then \phi it either a homomorphism or the negative of an antihomomorpliism of R onto R'. Theorem. Suppose that R is an associative ring and (R, +) does not contain element of the period 2 or 3,R' is a prime ring. If \phi is a 3-Lie isomorpJlism of R onto R'。then \phi is either a isomorphism or the negative of an anti-isomorphism of R onto R'.     

Elastic Analysis of Anisotropic Rotating Sandwich Circular Ring With a Functionally Graded Transition Region北大核心CSCD

The elastic analysis of anisotropic rotating sandwich ring with a functionally graded transition region was carried out. Like the shell sandwich structure in nature, the ring is composed of 3 well⁃bonded regions, of which the inner and outer regions are made of homogeneous anisotropic materials, and the intermediate transi⁃ tion region is made of a material with arbitrary⁃gradient properties along the radial direction. Based on the boundary conditions and the continuity conditions at the interface, the 2nd Fredholm integral equation for the radial stress was obtained with the integral equation method, then the stress and displacement fields of the sandwich ring structure were obtained through numerical solution. The distributions of the stress and displace⁃ ment fields in the sandwich ring structure were given. Different gradient changes encountered in engineering practice can be solved only through substitution of the corresponding function model. The effectiveness and ac⁃ curacy of the integral equation method were verified through comparison of the numerical solutions with the ex⁃ act ones for a special power function gradient variation form. The more general Voigt function model was adopt⁃ ed for the intermediate transition region, and the influences of the anisotropy degree, the gradient parameter, and the thickness on the stress and displacement fields were analyzed. The proposed Fredholm integral equation method provides a powerful tool for the optimal design of anisotropic functionally graded materials and sand⁃ wich ring structures. The numerical results make a theoretical guidance for the safety design of anisotropic functionally graded sandwich ring structures. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results.     

On Weakly Semicommutative Rings

A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicommutative rings and NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings.We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2-primal ring which satisfiesα-condition for an endomorphismαof R（that is,ab=0（？）aα（b）=0 where a,b∈R） then the skew polynomial ring R[x;α] is a weakly 2-primal ring,and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al.2007（Taiwanese J.Math.,11（5）（2007）, 1359-1368） considerably.Moreover,several new results about weakly semicommutative rings and NI-rings are included.     

几类加法非正则半环

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.     

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.     

关于超素环的左经典商环

In this paper,we define the left rank of a ring R and discuss the relatlons between a left ore reng R and its left classical quotient ring Q on the properties of their ideals and left ranks.Also we gave four equivalent types of the proposition:a ring R has a left classical quotient ring Q and Q is a division ring.As an application of this result,we gave a brief proof method which different from [2] on the theorem of the Goldie Prime rings.     

素环中的协中心化广义导子

Let R be a prime ring with center Z and S (?) R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x-xG(x) ∈ Z for all x ∈ S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime case is also considered.     

Some remarks on holomorphic functions and taylor series in C

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.