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. 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.