### Abstract

We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n_{0}, n_{1}, … , n_{k} are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).

Original language | English |
---|---|

Pages (from-to) | 833-852 |

Number of pages | 20 |

Journal | Monatshefte fur Mathematik |

Volume | 180 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 Aug 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Monatshefte fur Mathematik*, vol. 180, no. 4, pp. 833-852. https://doi.org/10.1007/s00605-015-0780-1

**An Engel condition with skew derivations for one-sided ideals.** / Liu, Cheng Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Engel condition with skew derivations for one-sided ideals

AU - Liu, Cheng Kai

PY - 2016/8/1

Y1 - 2016/8/1

N2 - We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).

AB - We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).

UR - http://www.scopus.com/inward/record.url?scp=84933566472&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84933566472&partnerID=8YFLogxK

U2 - 10.1007/s00605-015-0780-1

DO - 10.1007/s00605-015-0780-1

M3 - Article

AN - SCOPUS:84933566472

VL - 180

SP - 833

EP - 852

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 4

ER -