On Jordan Triple Derivations and Related Mappings

In this article we study certain functional equations and systems of functional equations related to (generalized) derivations on semiprime rings. In particular, we prove that any generalized Jordan triple derivation on a 2- torsion free semiprime ring is a generalized derivation. We also prove that any (generalized) Jordan triple *-derivation on a 2-torsion free semiprime *-ring is a (generalized) Jordan *-derivation. The second author was supported in part by the Ministry of Science, Education and Sports of the Republic of Croatia (project No. 037-0372784-2757).     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.     

Nil and strongly nil derivations

In a semiprime algebra R over a field of characteristic p ≥ 0, a derivation d is called nil if for each x ? R there is n such that dⁿx = 0. It is called strongly nil if it is induced by a nil element in the symmetric ring of quotients. The first result of this paper is an intrinsic criterion on a nil derivation being strongly nil. We then use this criterion to establish some relation between a nilpotent derivation and a strongly nilpotent derivation after a preliminary discussion of the nilpotency, or index of nilpotency, of a nilpotent derivation. Examples are given later in the paper to show that in some sense these results are best possible. This paper is self contained.     

环上的广义导子与Von Neumann代数上的P-核值保持映射

设Ａ是Ｂ（Ｈ）的子代数，ψ是Ａ到Ａ的线性映射，且对Ａ中的每个正交投影算子ｐ，有ψ（ｐ）（ｋｅｒｐ）ｒａｎｐ，则称ψ是Ａ到Ａ的Ｐ－核值保持映射，本文主要得到如下结果：每个２－非绕的半素环上的广义Ｊｏｒｄａｎ导子都是广义导子；每个ＶｏｎＮｅｕｍａｎｎ代数上的范数拓扑连续的Ｐ－核值保持映射是广义内导子．     

关于序半群的模糊拟理想

In this paper,fuzzy quasi-ideals of ordered semigroups are characterized by the properties of their level subsets.Furthermore,we introduce the notion of completely semiprime fuzzy quasi-ideals of ordered semigroups and characterize strongly regular ordered semigroups in terms of completely semiprime fuzzy quasi-ideals.Finally,we investigate the characterizations and decompositions of left and right simple ordered semigroups by means of fuzzy quasi-ideals.     

On the construction of involutory rhotrices

An involutory matrix is a matrix that is its own inverse. Such matrices are of great importance in matrix theory and algebraic cryptography. In this note, we extend this involution to rhotrices and present their properties. We have also provided a method of constructing involutory rhotrices.     

Generating functions for the powers of Fibonacci sequences

Whether or not there is an interaction between two factors in their effects on a dependent variable is often a central question. This paper proposes a general mechanism by which an interaction may arise: (a) the two factors are the same thing — or, at least, have a dimension in common — in the sense that it is meaningful to add (or subtract) them; (b) the sum of them (or the difference between them) is what determines the dependent variable; and (c) the relation between the sum (or difference) and the dependent variable is nonlinear. For example, if several factors contribute to arousal in an additive manner, and the relationship of performance score to arousal is inverted-U, the factors will appear to interact in their joint effect on performance. Psychological, medical, and other scientists are likely to be unfamiliar with the (nonlinear) equations used to express this type of theory. Consequently, the task of promoting and interpreting such ideas will fall to the mathematician and statistician.     

Prime and semiprime rings with symmetric skew 3-derivations

In this paper we introduce the notion of symmetric skew 3-derivations of prime or semiprime rings and prove that under certain conditions a prime ring with a nonzero symmetric skew 3-derivation has to be commutative.     

COMPARISON OF PRODUCTS OF THE VARIETIES OF LIE ALGEBRAS

Abstract

In Fortes (2001), we introduced a notion of order for associative pairs and we obtained a Goldie-like characterization of left orders in a semiprime pair coinciding with its socle. In this paper, we take up again that notion of order to establish a Faith-Utumi theorem, which studies left orders in a prime pair coinciding with its socle.     

弱半素子模的一些刻划

引入n′-系的概念,且给出弱半素子模的两个等价条件:(i)设K是左R-模M的子模,则K是M的弱半素子模当且仅当C(K)=M\K是n′-系;(ii)设K是左R-模M的子模,则K是M的弱半素子模当且仅当对任意f∈HomR(R,M),及任意A R,若f(A2)K,就有f(A)K.     

On generalized left derivations in rings and Banach algebras

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.     

Generalized Jordan left derivations on semiprime algebras

Let A{\mathcal{A}} be a semiprime algebra of characteristic not 2. Then any generalized Jordan left derivation on A{\mathcal{A}} is a generalized left derivation and is also a generalized derivation. This gives an affirmative answer to a question in Ashraf and Ali (Bull Korean Math Soc 45:253–261, 2008). Moreover, we prove that there are no nonzero generalized Jordan left derivations that take only nilpotent values on A{\mathcal{A}} .     

Fountain-Gould Left Orders for Associative Pairs

In this paper, we introduce a notion of weak Fountain-Gould left order for associative pairs and give a Goldie-like theory of associative pairs which are weak Fountain Gould left orders in semiprime pairs coinciding with their socles.     

环的广义斜导子

设R是一个半素环, RF(resp．Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上．设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0．     

On N1-free modules with trivial dual

In this paper we generalize the notion of Fountain-Gould order to the context of associative pairs. We give theorems about the existence and uniqueness of the left quotient pair and we obtain a Goldie-like haracterization which describes left orders in semiprime pairs coinciding with their socle.     

On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Bre?ar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) ? d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.     

A NOTE ON ADDITIVE RANK FUNCTIONS

Additive rank functions have been studied for Noetherian rings by Krause. It is shown that the notion of an additive rank function can be extended to more general classes of rings, and can be used in the characterization of semiprime Goldie rings and of orders in Artinian rings.     

Algebraic derivations with constants satisfying a polynomial identity

We prove that if a semiprime ringR possesses a derivation which is integral over its extended centroidC and whose constants satisfy a polynomial identity, thenR itself is a PI-ring. This answers affirmatively a problem raised by M. Smith in 1975 and recently again by Bergen and Grzeszczuk [4].     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

关于绝对半素环和绝对半素根

本文了绝对半素环的概念，研究了绝对半素与与半素环及素环之间的关系，讨论了由全体绝对半素环类决定的上根－－绝对半半素根及其性质。