### 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 ⋯ [(x^{n0} ); 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 |
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Pages (from-to) | 584-592 |

Number of pages | 9 |

Journal | Canadian Mathematical Bulletin |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Sep 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Canadian Mathematical Bulletin*,

*56*(3), 584-592. https://doi.org/10.4153/CMB-2011-185-5

}

*Canadian Mathematical Bulletin*, vol. 56, no. 3, pp. 584-592. https://doi.org/10.4153/CMB-2011-185-5

**On automorphisms and commutativity in semiprime rings.** / Liau, Pao Kuei; Liu, Cheng-Kai.

Research output: Contribution to journal › Article

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.

UR - http://www.scopus.com/inward/record.url?scp=84880960187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880960187&partnerID=8YFLogxK

U2 - 10.4153/CMB-2011-185-5

DO - 10.4153/CMB-2011-185-5

M3 - Article

AN - SCOPUS:84880960187

VL - 56

SP - 584

EP - 592

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 3

ER -