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Additive n-commuting maps on semiprime rings

Part of: Rings with polynomial identity Radicals and radical properties of rings

Published online by Cambridge University Press:  11 November 2019

Cheng-Kai Liu
Cheng-Kai Liu*
Affiliation:
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan, Republic of China ([email protected])
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Abstract

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

Keywords

commuting mappolynomial identityfunctional identity(semi)prime ringC*-algebraorthogonal completion

MSC classification

Primary: 16N60: Prime and semiprime rings
Secondary: 16R20: Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16R60: Functional identities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 193 - 216
Copyright
Copyright © Edinburgh Mathematical Society 2019

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