International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 120251 | 7 pages | https://doi.org/10.5402/2012/120251

On Generalized ( )-Derivations in Semiprime Rings

Academic Editor: C. Munuera
Received18 Oct 2012
Accepted06 Nov 2012
Published03 Dec 2012

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., [18]; 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 [917]).

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. 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. 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 © 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.

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