### Articles

Kyungpook Mathematical Journal 2018; 58(3): 481-488

**Published online** September 30, 2018

Copyright © Kyungpook Mathematical Journal.

### Generalized Derivations on *-prime Rings

Mohammad Ashraf, Malik Rashid Jamal^{*}

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

e-mail : mashraf80@hotmail.com

Department of Mathematics, Integral University, Lucknow-226026, India

e-mail : rashidmaths@gmail.com

**Received**: December 30, 2015; **Revised**: August 17, 2018; **Accepted**: August 20, 2018

### Abstract

Let

**Keywords**: derivation, generalized derivation, *-prime ring, centralizer.

### 1. Introduction

Let

* * (

x ) =x ,* (

x +y ) = * (x ) + * (y ) and* (

xy ) = * (y ) * (x )

hold for all

### 2. Identities Related to Generalized Derivations on *-ideals of *-prime Rings

Over the last three decades, several authors have explored various identities involving automorphisms or derivations on an appropriate subset of a prime or semiprime ring (see [1, 2, 3, 4, 5], where further references can be found). The purpose of this section is to prove some results which are of independent interest and related to generalized derivations on *-prime rings. We begin this section with the following well known result which are needed for developing the proof of the results presented in this section.

### Lemma 2.1

**Proof**

Given that

### Theorem 2.2

_{1}, _{2} : _{1}, _{2} : _{1}_{1} ≠ 0 _{2}, _{1}. _{1}(_{2}(_{1}(_{2}(_{2} = 0.

**Proof**

Replace _{1}(_{2}(

Replacing _{1}(

Now, replacing

Also, replacing

Using (

Since *-primeness of

Since _{1}, we get

Also, _{1}(_{2}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(

Replacing _{1}(_{1}, we find that [_{1}(_{1}) + [_{1}]_{1}(_{1} ∈ _{1}(_{1})_{1}(_{1}) = − [_{1}]_{1}(_{1}(_{1}) = {0}. Hence, we get [_{1}(_{1}) = 0 for all _{1} ∈ _{1}(_{1}) = 0. But [_{1}(_{1}) = 0 implies _{1} = 0 which is again a contradiction to the assumption that _{1}_{1} ≠ 0 .

Now, take _{2}(_{1}(

Replacing _{1}(

Replace _{2}(_{1}(_{1}(_{2}, we get *(_{2}(_{1}(_{1}(_{2} = 0 or _{1}_{1} = 0 on _{2} = 0 on _{2} = 0 on

Replace _{1}(_{2}(

Now, replacing _{1}(

Replace _{1}(_{2}(_{1}(_{2} = 0.

### Theorem 2.3

_{1}, _{2} : _{1}, _{2} : _{1}_{1} ≠ 0 _{2}, _{1}. _{1}(_{2}(_{1}(_{2}(_{2} = 0.

**Proof**

It is given that _{1}(_{2}(

Application of given condition yields that

Replacing _{1}(

Replacing

Arguing with similar lines as used after (

Take [_{1}(_{2}(

### Theorem 2.4

_{1}, _{2} : _{1}, _{2} : _{2}, _{1}. _{1}(_{2}(_{1} = 0 _{2} = 0.

**Proof**

Replace _{1}(_{2}(

On commuting with

Using the given hypothesis we obtain

Replacing _{1}(

Application of (

Replacing

Application of (

As a *-prime ring is also semiprime, (_{1}(_{1}(_{2}(_{2} and

Since _{1}(_{1}(_{2}(_{2} = 0 on _{2} = 0 on _{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}, we find that [_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1} = 0 on _{1} = 0 on _{1} = 0 on _{1}(_{1}(

Again replacing _{1}(_{1}(_{1}(_{1}(_{1} = 0 on

### Theorem 2.5

_{1}, _{2} : _{1}, _{2} : _{1}(_{2}(_{1} = _{2} = 0.

**Proof**

Let _{1}(_{2}(_{1}(_{2}(

Using the given hypothesis, we get

Replace _{1}(_{2}(_{1}(_{1}(_{1}(_{1}(_{1} = 0 on _{1} = 0 on _{1} = 0 on

If we replace _{1}(_{2}(_{2} = 0.

Now, take _{1}(_{2}(

### Theorem 2.6

_{1}, _{2} : _{1}, _{2} : _{1}(_{2}(_{1} = _{2} = 0.

**Proof**

Replacing _{1}(_{2}(_{1}(_{1}(_{1}(

Replacing _{1}(_{1}(_{1} = 0 on _{1}(_{2}(

Replacing _{2}(_{2}(_{2} = 0 on _{1}(_{2}(

### Theorem 2.7

_{1}, _{2} : _{1}, _{2} : _{1}. _{1}(_{2}(_{1}(_{2}(_{1} = 0 _{2} = 0.

**Proof**

Let _{1}(_{2}(_{1}(_{2}(_{1}(_{2}(_{1}(_{2}(

Using the given hypothesis, we find that

This implies that

Replacing _{2}(_{2}(_{1}(_{2}(_{1}(_{1} = 0 on _{1} = 0 on _{2}(_{2}(_{2}(_{2}(_{2}(_{2} = 0 on _{2} = 0 on

### Theorem 2.8

_{1}, _{2} : _{1}, _{2} : _{1}(_{2}(_{1} = _{2} = 0.

**Proof**

It is given that _{1}(_{2}(

Using the given condition, we get

Replacing _{1}(_{2}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1}(_{1} = 0 on _{1} = 0 on

Similarly, we can show that _{2} = 0.

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