# On automorphisms and commutativity in semiprime rings

Pao Kuei Liau, Cheng-Kai Liu

Research output: Contribution to journalArticle

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 584-592 9 Canadian Mathematical Bulletin 56 3 https://doi.org/10.4153/CMB-2011-185-5 Published - 2013 Sep 1

Semiprime Ring
Prime Ring
Commutativity
Commutator
Automorphism
Automorphisms
Denote
Integer

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

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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.",
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In: Canadian Mathematical Bulletin, Vol. 56, No. 3, 01.09.2013, p. 584-592.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On automorphisms and commutativity in semiprime rings

AU - Liau, Pao Kuei

AU - Liu, Cheng-Kai

PY - 2013/9/1

Y1 - 2013/9/1

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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