Kyungpook Mathematical Journal 2023; 63(3): 325-331
Published online September 30, 2023
Copyright © Kyungpook Mathematical Journal.
Annihilating Conditions of Generalized Skew Derivations on Lie Ideals
Nadeem ur Rehman and Sajad Ahmad Pary, Junaid Nisar∗
Department of Mathematics, Aligarh Muslim University, 202002 Aligarh, India
e-mail : email@example.com and firstname.lastname@example.org
Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Pune 412115, India
e-mail : email@example.com; firstname.lastname@example.org
Received: March 29, 2022; Revised: August 7, 2022; Accepted: August 22, 2022
Keywords: Prime ring, Generalized skew derivation
Throughout this article,
In , Sharma and Dhara proved that if
a =0or if .
a = 0or if and .
In , Dhara and De Filippis considered generalized derivations. They proved for a prime ring
In , Du and Wang demonstrated the following result for generalized derivations. Let
Inspired by the above outcomes, in the present paper, we prove the following result about generalized skew derivations with an annihilating condition on the non-central Lie ideal.
Theorem 1.1. Let
The following facts are often referenced in the proofs of our results:
Fact 2.1. Let
Fact 2.2. ([14, Lemma 2.1]) Let
is commutative or embeds in , for Fa field.
is algebraic of bounded degree 2 over .
Fact 2.3. ([7, Theorem 1]) Let
Fact 2.4. Let
Fact 2.5. ([2, Lemma 7.1]) Let
3. Some Important Results
We start with the following lemma and proposition; they are required for the development of our theorems:
Lemma 3.1. Suppose
In other words, we have
To complete the proof, suppose
Lemma 3.2. Let
Proposition 3.3. Let
4. Case of inner generalized skew derivation
In this case, we have
Lemma 4.1. Let
Lemma 4.2. Let
Proof of Theorem 1.1. In view of the Fact 2.4, a nonzero two-sided ideal
Case 1: If
Subcase 1: When α is an identity map, then the conclusion follows from Proposition 3.3.
Subcase 2: When α is inner , then there is
Subcase 3: When α is outer, then the conclusion follows from Lemma 4.2.
Case 2: When
This implies that
The authors are greatly indebted to the referee for his/her constructive comments and suggestions, which improved the quality of the paper.
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