Generalized derivations with Engel condition on Lie ideals of prime rings

Mohammad Aslam Siddeeque; Ali Ahmed Abdullah; Nazim Khan

doi:10.1515/gmj-2022-2190

Article

Generalized derivations with Engel condition on Lie ideals of prime rings

Mohammad Aslam Siddeeque , Ali Ahmed Abdullah and Nazim Khan

Published/Copyright: October 26, 2022

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From the journal Georgian Mathematical Journal Volume 30 Issue 1

Abstract

Consider ℜ as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 𝒵 ⁢ ( ℜ ) is the center of ℜ and 𝒬 is the Utumi ring of quotients where 𝒞 is the extended centroid of ℜ . Suppose 𝒫 to be a Lie ideal of ℜ which is non-central. Let 𝒦 be a generalized derivation of ℜ related with derivation μ of ℜ . If 𝒦 satisfies certain typical algebraic identities, then we prove that 𝒦 is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ embeds in M 2 ⁢ ( 𝒞 ) , a matrix ring of order 2 × 2 over 𝒞 .

Keywords: Prime ring; generalized derivation; Lie ideal; Utumi ring of quotients; generalized polynomial identities (GPIs); polynomial identities (PIs)

MSC 2010: 16W25; 16N60; 16R50 16R60

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Received: 2022-01-04

Revised: 2022-06-08

Accepted: 2022-06-13

Published Online: 2022-10-26

Published in Print: 2023-02-01

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Articles in the same Issue

https://doi.org/10.1515/gmj-2022-2190

Keywords for this article

Prime ring; generalized derivation; Lie ideal; Utumi ring of quotients; generalized polynomial identities (GPIs); polynomial identities (PIs)