### Abstract

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[...[[σ(x ^{n0}), x^{n1}], x^{n2}], ...], x^{nk}] = 0 for all x ∈ L, where n_{0}, n_{1}, n_{2}, ..., n _{k} are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

Original language | English |
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Article number | 1350092 |

Journal | Journal of Algebra and its Applications |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Mar 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Applied Mathematics

### Cite this

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*Journal of Algebra and its Applications*, vol. 13, no. 2, 1350092. https://doi.org/10.1142/S0219498813500928

**An engel condition with automorphisms for left ideals.** / Liu, Cheng-Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An engel condition with automorphisms for left ideals

AU - Liu, Cheng-Kai

PY - 2014/3/1

Y1 - 2014/3/1

N2 - Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[...[[σ(x n0), xn1], xn2], ...], xnk] = 0 for all x ∈ L, where n0, n1, n2, ..., n k are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

AB - Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[...[[σ(x n0), xn1], xn2], ...], xnk] = 0 for all x ∈ L, where n0, n1, n2, ..., n k are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

UR - http://www.scopus.com/inward/record.url?scp=84887409854&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887409854&partnerID=8YFLogxK

U2 - 10.1142/S0219498813500928

DO - 10.1142/S0219498813500928

M3 - Article

AN - SCOPUS:84887409854

VL - 13

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

SN - 0219-4988

IS - 2

M1 - 1350092

ER -