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 , .