Research Article | Open Access
Generalized Derivations and Left Ideals in Prime and Semiprime Rings
Let be an associative ring, a nonzero left ideal of , a derivation and a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) ; (2) ; (3) ; for all and .
Throughout this paper, let be an associative ring, a left ideal of , a derivation of and a generalized derivation of . For any two elements , will denote the commutator element and denotes . We use extensively the following basic commutator identities: and . Recall that a ring is called prime, if for any , implies that either or and is called semiprime if for any , implies . An additive mapping is said to be a derivation of if for any , holds. The generalized derivation of is defined as an additive mapping such that holds for any , where is a derivation of . So, every derivation is a generalized derivation, but the converse is not true in general. If , then we have for all , which is called a left multiplier mapping of . Thus, generalized derivation generalizes both the concepts, derivation as well as left multiplier mapping of .
In , Daif and Bell proved that if is a semiprime ring with a nonzero ideal and is a derivation of such that for all , then is central ideal. In particular, if , then is commutative. Recently, Quadri et al.  have generalized this result replacing derivation with a generalized derivation in a prime ring . More precisely, they obtained the following result.
Let be a prime ring and a nonzero ideal of . If admits a generalized derivation associated with a nonzero derivation such that any one of the following holds: (i) for all ; (ii) for all ; (iii) for all ; (iv) for all ; then is commutative.
Recently in , the first author of this paper has studied all the results of  in semiprime ring. In the present paper, our aim is to discuss similar identities in a left sided ideal of a semiprime rings.
2. Main Results
Theorem 2.1. Let be a semiprime ring and a nonzero left ideal of . If is a generalized derivation of associated with a derivation of such that for all , where , then .
Proof. If , then for any ,
that is, . This gives our conclusion. So let . Then by our assumption, we have
for all . Putting , , we obtain that . Since is a generalized derivation of , this implies that . This gives by using (2.2) that for all . Now, we replace with , where , and then we get
for all . Since is a left ideal, it follows that for all . Since is semiprime, it must contain a family of prime ideals such that . If is typical member of and , we have either or .
For fixed , the sets and form two additive subgroups of such that . Therefore, either or , that is, either or . Both of these two conditions together imply that for any . Therefore, .
Corollary 2.2. Let be a prime ring and be a nonzero left ideal of . If admits a generalized derivation associated with a derivation such that for all , where , then one of the following holds: (i);(ii) is commutative ring with ;(iii) is commutative ring with and for all .
Proof. By Theorem 2.1, we have . This gives Since is prime, either or . Now gives our conclusion (i). So, let which gives implying . Again, this gives . Since left annihilator of a left-sided ideal is zero, , that is, is commutative. If , we obtain conclusion (ii). So assume that . Then our assumption gives for all . Since , then for all . This gives for all , Let . Since is commutative, . Put in last result, we get . Since is commutative, using (2.5), it yields , implying . Then, (2.5) implies , which yields for all .
Theorem 2.3. Let be a semiprime ring, a nonzero left ideal of and a generalized derivation of associated with a derivation of . If for all , then .
Proof. If , then and hence, , which is our conclusion. Assume next that . Then by our assumption, we have for all . Put and get , that is, . Now using (2.6), the above relation yields for all . Putting , where , we obtain that , which is same as (2.3) in Theorem 2.1. By same argument of Theorem 2.1, we can conclude the result here.
Corollary 2.4. Let be a prime ring and a nonzero left ideal of . If is a generalized derivation of associated with a derivation of such that for all , where , then either is commutative or and one of the following holds: (i);(ii) for all . In case for all , with , then .
Proof. By the Theorem 2.3, we may conclude that . Then by same argument as given in Corollary 2.2, we obtain that either is commutative or . Let be noncommutative, then for any , we have , that is, acts as a left multiplier map on . Then for any , replacing with in our hypothesis , we have for all . Since acts as a left multiplier map on , this implies By using , it gives , that is, for all . Replacing with , where , we find that , which gives . Since is prime, either or for all . When , our assumption implies for all . This implies , that is, implies , unless .
Theorem 2.5. Let be a semiprime ring, a nonzero left ideal of and a derivation of . If for all , where , then . In case and is 2-torsion free, maps into its center.
Proof. We have for all , Putting , we get Using (2.9), we have Put in (2.11), and get Left multiplying (2.11) by and then subtracting from (2.12) yields for all . Put in (2.13), we have Left multiplying (2.13) by and then subtracting from (2.14), we get for all . This implies and hence, for all . Since is semiprime, . In case , for all , and then by , .
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Copyright © 2011 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.