### Article

Kyungpook Mathematical Journal 2022; 62(1): 43-55

**Published online** March 31, 2022

Copyright © Kyungpook Mathematical Journal.

### On *n*-skew Lie Products on Prime Rings with Involution

Shakir Ali and Muzibur Rahman Mozumder^{*}

epartment of Mathematics, Aligarh Muslim University, Aligarh-202002, U. P., India

e-mail : shakir.ali.mm@amu.ac.in and muzibamu81@gmail.com

**Received**: July 7, 2021; **Revised**: October 5, 2021; **Accepted**: October 7, 2021

Let

**Keywords**: Prime ring, derivation, involution, centralizing mappings, 2-skew Lie product, 2-skew centralizing mappings, *n*-skew commuting mappings, *n*-skew centralizing mapping

### 1. Introduction

This research is motivated by the recent work's of Ali-Dar [1], Qi-Zhang [5] and Hou-Wang [3]. However, our approach is different from that of the authors of [5] and [3]. A ring

For any ^{*}_{*}[x,f(x)]_{n}=0

The objective of this paper is to introduce the notion of

### 2. Main Results

In order to study the effect of

**Lemma 2.1.** Let

Substituting

Combining (2.1) and (2.2), we conclude that ^{*}

**Lemma 2.2.** Let

Replacing

By (2.3) and (2.4), we conclude that

Taking

Substitute

**Theorem 2.3.** Let

**Case**

Linearizing the above expression, we get

That is,

This further implies that

Hence

Replacing

Using the primeness of

First we consider the situation

Substituting

Since

Replacing

for all

**Case** (

On expansion we acquire

Replacing

Then by the primeness of

Substituting

Subtracting (2.10) from (2.11) and using 2-torsion freeness of

Application of (2.9) yields

Putting

Taking

**Theorem 2.4.** Let

**Case** (

Linearizing this, we get

This implies that

Replacing

Using the primeness of

for all

This implies that

Taking

**Case** (

On expansion we get

Linearization of (2.17) yields

Substituting

Since

Replacing

By the primeness of the ring

for all

This implies that

Substituting

Since

Combining this with (2.19) and using the fact that

Replacing

**Theorem 2.5.** Let

If we take

Replacing

This is same as the relation (2.10) in Theorem 2.3 and hence we conclude that

Linearization of (2.24) gives

The primeness of

**Theorem 2.6.** Let _{1}_{2}

If either _{1}_{2}_{1}, d_{2}

for all

This implies either

Since

Linearizing (2.28), we obtain

Replacing

for all

for all

Replacing

**Theorem 2.7.** Let _{1}_{2}

_{1}_{2}

If _{2}_{1}

Replacing

Now consider the second case in which both _{1}_{2}

for all

By the primeness of the ring

Replacing

for all

for all

for all

Subtracting (2.38) form (2.37), we get

Substituting _{1}(k)k=0_{1}(k)=0

Replacing

As an immediate consequence of the above theorem, we get the following corollary:

**Corollary 2.8.** Let

The following example shows that the second kind involution assumption is essential in Theorem 2.3 and Theorem 2.4.

**Example 2.9.** Let

Obviously, _{1}_{2}

**Example 2.10.** Let _{1}_{2}_{1}_{2}_{1}, D_{2}

We conclude the paper with the following Conjectures.

**Conjecture 2.11.** Let

**Conjecture 2.12.** Let

**Conjecture 2.13.** Let

### Acknowledgments

The authors are greatful to the learned referee for carefully reading the manuscript. The valuable suggestions have simplified and clarified the paper greatly.

- S. Ali and N. A. Dar,
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Strong 2-skew commutativity preserving maps on prime rings with involution , Bull. Malays. Math. Sci. Soc.,42(1) (2019), 33-49. - E. C. Posner,
Derivations in prime rings , Proc. Amer. Math. Soc.,8 (1957), 1093-1100. - X. Qi and Y. Zhang,
k-skew Lie products on prime rings with involution , Comm. Algebra,46(3) (2018), 1001-1010.