Commuting Traces of Multiadditive Mappings

Lee, Pjek-Hwee; Wong, Tsai–Lien; Lin, Jer-Shyong; Wang, Ren-June

doi:10.1006/jabr.1996.7016

Cited by 44 publications

(15 citation statements)

References 5 publications

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“…When we investigate the above-mentioned mappings, the principal task is to describe their forms. This is demonstrated by various works, see [8,10,11,12,13,15,20,21,28,29,33,36,41,44,45,49,50]. We encourage the reader to read the well-written survey paper [13], in which the author presented the development of the theory of semi-centralizing mappings and their applications in details.…”

Section: Introductionmentioning

confidence: 84%

“…Moreover, every centralizing mapping of a prime ring of characteristic not 2 is of the proper form and is actually commuting. Lee et al further generalized Brešar's results by showing that each commuting trace of an arbitrary multilinear mapping on a prime ring also has the proper form [28]. Cheung in [21] studied commuting mappings of triangular algebras (e.g., of upper triangular matrix algebras and nest algebras).…”

Section: Introductionmentioning

confidence: 96%

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Centralizing traces and Lie triple isomorphisms on triangular algebras

Xiao

Wei

Fošner

2014

Linear and Multilinear Algebra

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Abstract. Let T be a triangular algebra over a commutative ring R and Z(T ) be the center of T . Suppose that q : T × T −→ T is an R-bilinear mapping and that Tq : : T −→ T is a trace of q. We describe the form of Tq satisfying the condition [Tq(T ), T ] ∈ Z(T ) for all T ∈ T . The question of when Tq has the proper form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on T to be almost standard. As applications we characterize Lie triple isomorphisms of triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 96%

Centralizing traces and Lie triple isomorphisms on triangular algebras

Xiao

Wei

Fošner

2014

Linear and Multilinear Algebra

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“…It has turned out that this study is closely related to the problem of characterizing Lie isomorphisms or Lie derivations of associative rings [7,13,21,22,41,75]. Lee et al further generalized Brešar's results by showing that each commuting trace of an arbitrary multilinear mapping on a prime ring also has the proper form [51]. On the other hand, centralizing mappings are looked on as the most basic and important examples of functional identities [21].…”

Section: Introductionmentioning

confidence: 99%

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Liang

Wei

Xiao

et al. 2014

Linear and Multilinear Algebra

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Abstract. Let G be a generalized matrix algebra over a commutative ring R and Z(G) be the center of G. Suppose that q : G × G −→ G is an R-bilinear mapping and Tq : G −→ G is the trace of q. We describe the form of Tq satisfying the condition [Tq(G), G] ∈ Z(G) for all G ∈ G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of G to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

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“…This theory has been developed over the last few years in [174,779,14,15,18722,25,27,34,35] . The goal is to describe either the form of the maps involved or the structure of the ring with these identities.…”

Section: Introductionmentioning

confidence: 99%

FUNCTIONAL IDENTITIES WITH r-INDEPENDENT COEFFICIENTS

Бейдар

Bresˇar

Chebotar

2002

Communications in Algebra

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derivation provided that dð½x; yÞ ¼ ½dðxÞ; y þ ½x; dðyÞ for all x; y 2 A. The study of Lie maps of associative rings goes back to Hua (1951). At his 1961 AMS Hour Talk [32, pp. 528-529] Herstein formulated a program of studying Lie maps in associative (prime or simple) rings. Over the next 30 years Martindale and some of his students solved a number of Herstein's problems in the class of prime rings under additional assumption on existence of certain idempotents (see [36] and references over there).In 1993 Bresˇar [19] solved Herstein's problem on the description of a Lie isomorphism a : A ! B of prime rings A and B without assuming the existence of idempotents via the solution of the following FI: ½Bðx; xÞ; x ¼ 0 for all x 2 A, where B : A Â A ! A is a biadditive map (in the case of Lie isomorphisms Bðx; yÞ ¼ a À1 ðaðxÞaðyÞ þ aðyÞaðxÞÞ). He also applied this FI to the description of commutativity preserving maps of prime rings (i.e., such maps a that ½x; y ¼ 0 implies ½aðxÞ; aðyÞ ¼ 0). Since then FIs became the fundamental tool in the solution of all remaining unsolved Herstein's problems. [5,6,11,12,15,17,19,26,28,43,44] Besides their role in the study of commutativity preserving maps, [1,7,19] FIs were successfully applied to the study of additive maps maps preserving n th powers, [24] additive maps preserving an arbitrary fixed multilinear polynomial of degree ! 2 [5,13] and the structure theory of certain nonassociative algebras. [10] On the one hand, recently a number of papers on the Lie structure of associative superalgebras [30,31,37,39] have been published. On the other hand, Herstein's problems on Lie maps in associative algebras have been solved. Finally, a question of nonexistence of isomorphisms of certain Lie superalgebras arising from the associative ones has been already posed by Montgomery. [39, p. 571] Therefore it is now natural to consider the analogues of Herstein's problems in the context of associative superalgebras. It has turned out that the current level of the theory of functional identities is not sufficient for solving these problems because the automorphism s defining the Z 2 -graduation on an associative prime superalgebra A is involved in the FIs arising from the study of Lie maps in A. In the case when s is X-inner, i.e., when there exists an invertible element t 2 Q s , the symmetric Martindale ring of A, such that t 2 is central in Q s , these FIs involves the element t. Because of that, the process of exclusion of variables, developed by Beidar in [2] , became essentially more complicated. We overcome this problem by introducing in Sec. 1 the concept of an r-independent set upon which this paper is based. In Sec. 2 we introduce and study the concept of a d-free subset in the context of FIs with d-independent coefficients. This concept is an analog of that of a d-free subset for FIs without coefficients introduced by Beidar and Chebotar in [8] and which, together with corresponding results, [8,9] played the crucial role in the solution of a number of Herstein's Lie map proble...

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Commuting Traces of Multiadditive Mappings

Cited by 44 publications

References 5 publications

Centralizing traces and Lie triple isomorphisms on triangular algebras

Centralizing traces and Lie triple isomorphisms on triangular algebras

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

FUNCTIONAL IDENTITIES WITH r-INDEPENDENT COEFFICIENTS

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