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# On Generalized ()-Derivations in Semiprime Rings

**Academic Editor:**C. Munuera

#### Abstract

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

#### 1. Introduction

Throughout the present paper, always denotes an associative semiprime ring with center . For any , the commutator and anticommutator of and are denoted by and and are defined by and , respectively. Recall that a ring is said to be prime, if for , implies either or and is said to be semiprime if for , implies . An additive mapping is said to be derivation if holds for all . The notion of derivation is extended to generalized derivation. The generalized derivation means an additive mapping associated with a derivation such that holds for all . Then every derivation is a generalized derivation, but the converse is not true in general.

A number of authors have studied the commutativity theorems in prime and semiprime rings admitting derivation and generalized derivation (see e.g., [1–8]; where further references can be found).

Let and be two endomorphisms of . For any , set and . An additive mapping is called a -derivation if holds for all . By this definition, every -derivation is a derivation, where means the identity map of . In the same manner the concept of generalized derivation is also extended to generalized -derivation as follows. An additive map is called a generalized -derivation if there exists a -derivation such that holds for all . Of course every generalized -derivation is a generalized derivation of , where denotes the identity map of .

There is also ongoing interest to study the commutativity in prime and semiprime rings with -derivations or generalized -derivations (see [9–17]).

The present paper is motivated by the results of [17]. In [17], Rehman et al. have discussed the commutativity of a prime ring on generalized -derivation, where and are automorphisms of . More precisely, they studied the following situations: (i) ; (ii) ; (iii) ; (iv) ; (v) for all , where is a nonzero ideal of .

The main objective of the present paper is to extend above results for generalized -derivations in semiprime ring , where and are considered as epimorphisms of .

To prove our theorems, we will frequently use the following basic identities:

#### 2. Main Results

Theorem 2.1. * Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *First we consider the case
for all . Replacing by in (2.1) we get
Using (2.1), it reduces to
for all . Again replacing by in (2.3), we get
for all and . Left multiplying (2.3) by and then subtracting from (2.4) we have
for all and . Replacing with , , we get for all and . Since is an epimorphism of , we can write
for all .

Since is semiprime, it must contain a family of prime ideals such that . If is a typical member of and , it follows that

Construct two additive subgroups and . Then . Since a group cannot be a union of two its proper subgroups, either or , that is, either or . Thus both cases together yield for any . Therefore, , that is, . Thus
and so . This implies , where is a nonzero ideal of , since . Then . Since is semiprime, it follows that , that is, .

Similarly, we can obtain the same conclusion when for all .

Theorem 2.2. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We begin with the case
for all . Replacing by in (2.9) we get
Right multiplying (2.9) by and then subtracting from (2.10) we get
for all .

Now replacing by in (2.11), we obtain
for all and for all . Left multiplying (2.11) by and then subtracting from (2.12), we get
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case for all .

Theorem 2.3. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We assume first that for all . This implies
Replacing by in (2.14) we have
Right multiplying (2.14) by and then subtracting from (2.15), we get
Now replacing by , where , in (2.16), we obtain
Left multiplying (2.16) by and then subtracting from (2.17), we get that
that is,
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case for all .

Theorem 2.4.

* Proof. *By our assumption first consider for all . This gives
Replacing by in (2.20), we have
Right multiplying (2.20) by and then subtracting from (2.21), we obtain that
Now replacing by , where , in (2.22) and by using (2.22), we obtain
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar argument can be adapted in case for all .

Theorem 2.5. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We begin with the situation
for all . Replacing by in (2.24), we get
Right multiplying (2.24) by and then subtracting from (2.25), we obtain that
for all . Now replacing by in (2.26), where , and by using (2.26), we obtain
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

In case for all , the similar argument can be adapted to draw the same conclusion.

We know the fact that if a prime ring contains a nonzero central ideal, then must be commutative (see Lemma 2 in [18]). Hence the following corollary is straightforward.

Corollary 2.6. *Let be a prime ring, and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of satisfying any one of the following conditions:*(1)* for all or for all ;*(2)* for all or for all ;*(3)* for all or for all ;*(4)* for all or for all ;*(5)* for all or for all ;**then must be commutative.*

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#### Copyright

Copyright © 2012 Basudeb Dhara and Atanu Pattanayak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.