On automorphisms and commutativity in semiprime rings

Pao Kuei Liau; Cheng Kai Liu

doi:10.4153/CMB-2011-185-5

On automorphisms and commutativity in semiprime rings

Pao Kuei Liau, Cheng Kai Liu

Department of Mathematics

Research output: Contribution to journal › Article

11 Citations (Scopus)

Abstract

Let R be a semiprime ring with center Z(R). For x; y 2 R, we denote by [x; y] = xy - yx the commutator of x and y. If is a non-identity automorphism of R such that ⋯ [(xⁿ⁰ ); xn1 ]; xn2 ; ⋯ ; xnk = 0 for all x 2 R, where n0; n1; n2; ⋯ ; nk are fixed positive integers, then there exists a map : R ! Z(R) such that (x) = x + (x) for all x 2 R. In particular, when R is a prime ring, R is commutative.

Original language	English
Pages (from-to)	584-592
Number of pages	9
Journal	Canadian Mathematical Bulletin
Volume	56
Issue number	3
DOIs	https://doi.org/10.4153/CMB-2011-185-5
Publication status	Published - 2013 Sep 1

All Science Journal Classification (ASJC) codes

Mathematics(all)

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Liau, P. K., & Liu, C. K. (2013). On automorphisms and commutativity in semiprime rings. Canadian Mathematical Bulletin, 56(3), 584-592. https://doi.org/10.4153/CMB-2011-185-5

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N2 - Let R be a semiprime ring with center Z(R). For x; y 2 R, we denote by [x; y] = xy - yx the commutator of x and y. If is a non-identity automorphism of R such that ⋯ [(xn0 ); xn1 ]; xn2 ; ⋯ ; xnk = 0 for all x 2 R, where n0; n1; n2; ⋯ ; nk are fixed positive integers, then there exists a map : R ! Z(R) such that (x) = x + (x) for all x 2 R. In particular, when R is a prime ring, R is commutative.

AB - Let R be a semiprime ring with center Z(R). For x; y 2 R, we denote by [x; y] = xy - yx the commutator of x and y. If is a non-identity automorphism of R such that ⋯ [(xn0 ); xn1 ]; xn2 ; ⋯ ; xnk = 0 for all x 2 R, where n0; n1; n2; ⋯ ; nk are fixed positive integers, then there exists a map : R ! Z(R) such that (x) = x + (x) for all x 2 R. In particular, when R is a prime ring, R is commutative.

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