Generalized Derivations in Semiprime Gamma Rings
Let be a 2-torsion-free semiprime -ring satisfying the condition for all , and let be an additive mapping such that for all and for some derivation of . We prove that is a generalized derivation.
Hvala  first introduced the generalized derivations in rings and obtained some remarkable results in classical rings. Daif and Tammam El-Sayiad  studied the generalized derivations in semiprime rings. The authors consider an additive mapping of a ring with the property for some derivation of . They prove that is a Jordan generalized derivation.
Aydin  studied generalized derivations of prime rings. The author proved that if is an ideal of a noncommutative prime ring , is a fixed element of and is an generalized derivation on associated with a derivation then the condition or for all implies .
Çeven and Öztürk  have dealt with Jordan generalized derivations in Γ-rings and they proved that every Jordan generalized derivation on some Γ-rings is a generalized derivation.
Generalized derivations of semiprime rings have been treated by Ali and Chaudhry . The authors proved that if is a commuting generalized derivation of a semiprime ring associated with a derivation then for all and is central. They characterized a decomposition of relative to the generalized derivations.
Atteya  proved that if is nonzero ideal of a semiprime ring and admits a generalized derivation such that then contains a nonzero central ideal.
In this paper, we prove the following results
Let be a 2-torsion-free semiprime Γ-ring satisfying the following assumption: and be an additive mapping. If there exists a derivation of such that for all , , then is a Jordan generalized derivation.
Let and Γ be additive abelian groups. is called a Γ-ring if there exists a mapping such that for all , the following conditions are satisfied:(i),(ii), , , .
For any and for the expressions is denoted by and are denoted by . Then one has the following identities: for all and for all . Using the assumption (*) the above identities reduce to for all and , .
Further stands for a prime Γ-ring with center . The ring is -torsion-free if , implies , where is a positive integer, is prime if implies or , and it is semiprime if implies . An additive mapping is called a left (right) centralizer if for , and it is called a Jordan left (right) centralizer if A mapping is called biadditive if it is additive in both arguments. An additive mapping is called a derivation if for all , and it is called a Jordan derivation if for all , . A derivation is inner if there exists , such that holds for all , . Every derivation is a Jordan derivation. The converse is in general not true. An additive mapping is said to be a generalized derivation if there exists a derivation such that for all , . The maps of the form where , are fixed elements in and for all called the generalized inner derivation. An additive mapping is said to be a Jordan generalized derivation if there exists a derivation such that for all , . Hence the concept of a generalized derivation covers both the concepts of a derivation and a left centralizers and the concept of a Jordan generalized derivation covers both the concepts of a Jordan derivation and a left Jordan centralizers. An example of a generalized derivation and a Jordan generalized derivation is given in .
3. Main Results
We start from the following subsidiary results.
Lemma 3.1. Let be a semiprime Γ-ring. If are such that for all , , then .
Proof. Let . Then By semiprimeness of with respect to , it follows that .
Lemma 3.2. Let be a semiprime Γ-ring and biadditive mappings. If for all , then for all , .
Proof. First we replace with in the relation , and use the biadditivity of the and . Then we have
Hence by semiprimeness of with respect to .
Now we replace by and obtain the assertion of the lemma with the similar observation as above.
Lemma 3.3. Let be a semiprime Γ-ring satisfying the assumption (*) and be a fixed element of . If for all , , then .
Proof. First we calculate the following expressions using the assumption (*), Since for all , , we get . By the semiprimeness of we get for all . Hence .
Lemma 3.4. Let be a Γ-ring satisfying the condition (*) and be a Jordan generalized derivation with the associated derivation . Let and . Then(i). In particular, if is 2-torsion-free, then(ii),(iii).
Proof. (i) We have , for all , . Then replacing by , and following the series of implications below we get the result:
(ii) Replace by in the above relation (3.5), then we get, Using the assumption (*), we conclude that Again, replacing by in (3.5) Adding both sides , we get, Comparing (3.7) and (3.9) we obtain, Since is 2-torsion-free, it gives
(iii) Replace for in (3.12), we get,
Definition 3.5. Let be a Γ-ring and be a Jordan generalized derivation with the associated derivation . Define for all and .
Lemma 3.6. The function has the following properties:(i). (ii). (iii). (iv).
Proof. The results easily follow from Lemma 3.4(i).
Remark 3.7. is a generalized derivation if and only if for all , .
Theorem 3.8. Let be a 2-torsion-free semiprime Γ-ring satisfying the condition (*). Let and . Then(i) for all ,(ii).
Lemma 3.9. , for all , .
Theorem 3.10. Let be a 2-torsion-free semiprime Γ-ring satisfying the assumption (*) and be a Jordan generalized derivation with associated derivation on . Then is a generalized derivation.
Proof. In particular, , for all , . This gives
Then . Now we will compute each side of this equality by using (3.15) and the above relation,
So we get
So we get
Comparing the results of (3.22) and (3.24) and using the above relations
where stands for .
On the other hand, we have Now we use (3.15) and the properties of , to derive which gives Moreover, So we obtain Comparing (3.31) and (3.33), we derive Finally using (3.31) we get . But . By the semiprimeness of , we have . Again by the primeness of , we get . The proof is complete.
It is clear that if we let the derivation to be the zero derivation in the above theorem, we get the following result.
Theorem 3.11. Let be a 2-torsion-free semiprime Γ-ring and be an additive mapping which satisfies for all , . Then is a left centralizer
Proof. We have
If we replace by , we get
By replacing with and using (3.5), we arrive at
But on the other hand,
Comparing (3.37) and (3.38) we obtain
If we linearize (3.39) in , we get
Now we shall compute in two different ways. If we use (3.39) we have
But if we use (3.40) we have
Comparing (3.41) and (3.42) and introducing a bi-additive mapping we arrive at
Equality (3.36) can be rewritten in this notation as . Using this fact and (3.43) we obtain
Using first Lemma 3.2 and then Lemma 3.1 we have
Now fix some and using Lemma 3.3 we get .
In particular, , for all . This gives Therefore . Both sides of this equality will be computed in few steps using (3.36), Since , we obtain On the other hand, we also have