### Articles

Kyungpook Mathematical Journal 2019; 59(1): 1-11

**Published online** March 23, 2019

Copyright © Kyungpook Mathematical Journal.

### Study of Generalized Derivations in Rings with Involution

Muzibur Rahman Mozumder and Adnan Abbasi, Nadeem Ahmad Dar^{*}

Department of Mathematics, Aligarh Muslim University, Aligarh, India, e-mail: muzibamu81@gmail.com and adnan.abbasi001@gmail.com

Govt. HSS, Kaprin, Shopian, Jammu and Kashmir, India, e-mail: ndmdarlajurah@gmail.com

**Received**: March 20, 2018; **Revised**: August 8, 2018; **Accepted**: August 13, 2018

Let ^{*} ∈ ^{*}]) ± ^{*} ∈ ^{*}) ± [^{*}] ∈ ^{*}) ± ^{*} ∈ ^{*})] ± ^{*} ∈ ^{*} ∈ ^{*}] ∈ ^{*}] ? ^{*} ∈ ^{*}) ∈

**Keywords**: prime ring, generalized derivation, derivation, involution.

### 1. Introduction

Throughout this paper ^{*})^{*} = ^{*} = ^{*} = −^{*} = ^{*}

A derivation on

Very recently in many papers the additive mappings like derivations, generalized derivations have been studied in the setting of rings with involution and in fact it was seen that there is a close connection between these mappings and the commutativity of the ring ^{*}] ∈

### 2. Main Results

We begin our investigation with the following lemmas, which are essential to prove our main results.

### Lemma 2.1.([15])

^{*}] ∈

### Lemma 2.2.([15])

^{*} ∈

### Theorem 2.3

^{*} ∈

**Proof**

By the given hypothesis, we have

A Linearization of (

This can be further written as

Replacing

On solving, we obtain

and ^{*}, ^{*}, ^{*}, ^{*},

Using (^{*}, ^{*}, ^{*},

### Theorem 2.4

^{*}]) ± ^{*} ∈

**Proof**

We first consider the case

If ^{*} ∈

This can be further written as

Replacing

for all

+[^{*}, ^{*}], ^{*}], ^{*}], ^{*}], ^{*}], ^{*}], ^{*}] + [^{*}] ∈ ^{*}] ∈

and

for all ^{*}]) + ^{*})^{*}]) + ^{*}) ∈ ^{*}, we get (^{2} ∈ ^{2} ∈

The second case can be proved in a similar manner with necessary variations.

### Theorem 2.5

^{*}) ± [^{*}] ∈

**Proof**

We first consider the positive sign case. If ^{*}] ∈

Linearizing (

Replacing

Now using the primeness of ^{*} + ^{*}) ∈ ^{*} ∈ ^{*}) + [^{*}])^{*}) + [^{*}] ∈ ^{*}, we get ^{2}) ∈ ^{2}) ∈

### Theorem 2.6

^{*}) ± ^{*} ∈

**Proof**

We first consider the case

If either ^{*} ∈

Replacing

Using the primeness condition, we get either ^{*} + ^{*}) ∈ ^{*} +^{*}) ∈

Replacing ^{*}), ^{*}), ^{*} we have [

This further implies that ^{*}) + ^{*} ∈

Similarly we can prove the second part.

### Theorem 2.7

^{*})] ± ^{*} ∈

**Proof**

The proof is on the similar lines as in the above theorem.

### Theorem 2.8

^{*} ∈

**Proof**

we first consider the case

If ^{*} ∈

Replacing ^{*})^{*} ∈ ^{*} ∈

Similarly we can prove the other case.

On similar lines we can prove the following result.

### Theorem 2.9

^{*}] ∈

### Theorem 2.10

^{*}] ∓ ^{*} ∈

**Proof**

We first consider the case

Linearizing (

Replacing

Using the primeness condition we have [^{*}] − ^{*} ∈ ^{*}] − ^{*} ∈ ^{2} ∈

Since

Replacing ^{*}, we get

This can be further written as

Replacing

Replacing ^{2} +2^{2}^{2}[^{2}

Replacing

Now we consider the case

Using the same steps as we did in the above case, we get

Replacing ^{*}, we have

This can be further written as

That is, 2

Replacing

Using (

Replacing

Using the primeness and the fact that

### Theorem 2.11

^{*}) ∈

**Proof**

Proceeding on the similar lines as we did in the previous result, we get ^{*}) ∈ ^{*} where

At the end of paper, let us write an example which shows that the restriction of the second kind involution in our results is not superfluous

### Example 2.21

Let

Obviously,

Then ^{*} =

F (x ) ∘x ^{*}∈Z (R ),F ([x, x ^{*}]) ±x ∘x ^{*}∈Z (R ),F (x ∘x ^{*}) ± [x, x ^{*}] ∈Z (R ),F (x ) ∘D (x ^{*}) ±x ∘x ^{*}∈Z (R ),[

F (x ),D (x ^{*})] ±x ∘x ^{*}∈Z (R ),F (x ) ±x ∘x ^{*}∈Z (R ),F (x ) ± [x, x ^{*}] ∈Z (R ),[

F (x ),x ^{*}] ∓F (x ) ∘x ^{*}∈Z (R ),F (x ∘x ^{*}) ∈Z (R ),

for all

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