k-commuting mappings of generalized matrix algebras

Abstract

Motivated by the intensive and powerful works of Beidar [1,4] and Brešar [8,12], we will study k-commuting mappings of generalized matrix algebras in this article. The general form of arbitrary k-commuting mapping of a generalized matrix algebra is determined. It is shown that under suitable hypotheses, every k-commuting mapping of a generalized matrix algebra take a certain form which is said to be proper. A number of applications related to k-commuting mappings are presented. These results not only give new perspectives to the aforementioned works of Beidar and Bresar but also extend the main results of Cheung, Du and Wang [17,19] to the case of generalized matrix algebras.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

“…Let M n (R) be the full matrix algebra consisting of all n × n matrices over R. It is worth to point out that the notion of generalized matrix algebras efficiently unifies triangular algebras with full matrix algebras together. The feature of our systematic work is to deal with all questions related to (non-)linear mappings of triangular algebras and of full matrix algebras under a unified frame, which is the admired generalized matrix algebras frame, see [40], [41], [42], [61], [62], [63].…”

“…It was Krylov who initiated the study of linear mappings on generalized matrix algebras from the classifying point of view [34]. Since then many articles are devoted to this topic, and a number of interesting results are obtained (see [1], [3], [6], [7], [12], [19], [23], [24], [40], [41], [42], [60], [61], [62]). Nevertheless, it leaves so much to be desired.…”

Automorphisms of Formal Matrix Rings

“…Automorphisms and isomorphisms of various matrix rings are studied in many papers; for example, see [1], [3], [4], [10], [11], [12], [13], [17], [18], [20], [22], [26], [27]. Some other mappings of matrix rings were also studied; in particular, commuting and centralizing mappings were studied (for example, see [24]). The author's work [20] is devoted to automorphisms and homomorphsms of formal matrix algebras.…”

Modules over Discrete Valuation Rings

“…For rings of generalized matrices, there are studies of their commuting, self-centralizing and other close mapings, as well as a number of related issues (see [40], [41], [51]). It may be interesting and advisable to study similar maps of incidence rings.…”