Generalized Derivations in Semiprime Gamma Rings

Abstract

Let 𝑀 be a 2-torsion-free semiprime Γ-ring satisfying the condition 𝑎𝛼𝑏𝛽𝑐=𝑎𝛽𝑏𝛼𝑐 for all 𝑎,𝑏,𝑐∈𝑀,𝛼,𝛽∈Γ, and let 𝐷∶𝑀→𝑀 be an additive mapping such that 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥+𝑥𝛼𝑑(𝑥) for all 𝑥∈𝑀,𝛼∈Γ and for some derivation 𝑑 of 𝑀. We prove that 𝐷 is a generalized derivation.

1. Introduction

Hvala [1] first introduced the generalized derivations in rings and obtained some remarkable results in classical rings. Daif and Tammam El-Sayiad [2] studied the generalized derivations in semiprime rings. The authors consider an additive mapping 𝐺∶𝑅→𝑅 of a ring 𝑅 with the property 𝐺(𝑥²)=𝐺(𝑥)𝑥+𝑥𝐷(𝑥) for some derivation 𝐷 of 𝑅. They prove that 𝐺 is a Jordan generalized derivation.

Aydin [3] studied generalized derivations of prime rings. The author proved that if 𝐼 is an ideal of a noncommutative prime ring 𝑅, 𝑎 is a fixed element of 𝑅 and 𝐹 is an generalized derivation on 𝑅 associated with a derivation 𝑑 then the condition 𝐹([𝑥,𝑎])=0 or [𝐹(𝑥),𝑎]=0 for all 𝑥∈𝐼 implies 𝑑(𝑥)=𝜆[𝑥,𝑎].

Çeven and Öztürk [4] 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 [5]. The authors proved that if 𝐹 is a commuting generalized derivation of a semiprime ring 𝑅 associated with a derivation 𝑑 then 𝑑(𝑥)[𝑦,𝑧]=0 for all 𝑥,𝑦,𝑧∈𝑅 and 𝑑 is central. They characterized a decomposition of 𝑅 relative to the generalized derivations.

Atteya [6] proved that if 𝑈 is nonzero ideal of a semiprime ring 𝑅 and 𝑅 admits a generalized derivation 𝐷 such that 𝐷(𝑥𝑦)−𝑥𝑦∈𝑍(𝑅) then 𝑅 contains a nonzero central ideal.

Rehman [7, 8] studied the commutativity of a ring 𝑅 by means of generalized derivations acting as homomorphisms and antihomomorphisms.

In this paper, we prove the following results

Let 𝑀 be a 2-torsion-free semiprime Γ-ring satisfying the following assumption:𝑎𝛼𝑏𝛽𝑐=𝑎𝛽𝑏𝛼𝑐∀𝑎,𝑏,𝑐∈𝑀,𝛼,𝛽∈Γ, (∗) and 𝐷∶𝑀→𝑀 be an additive mapping. If there exists a derivation 𝑑 of 𝑀 such that 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥+𝑥𝛼𝑑(𝑥) for all 𝑥∈𝑀, 𝛼∈Γ, then 𝐷 is a Jordan generalized derivation.

2. Preliminaries

Let 𝑀 and Γ be additive abelian groups. 𝑀 is called a Γ-ring if there exists a mapping 𝑀×𝑀×𝑀→𝑀 such that for all 𝑎,𝑏,𝑐∈𝑀, 𝛼,𝛽∈Γ the following conditions are satisfied:(i)𝑎𝛽𝑏∈𝑀,(ii)(𝑎+𝑏)𝛼𝑐=𝑎𝛼𝑐+𝑏𝛼𝑐, 𝑎(𝛼+𝛽)𝑏=𝑎𝛼𝑏+𝑎𝛽𝑏, 𝑎𝛼(𝑏+𝑐)=𝑎𝛼𝑏+𝑎𝛼𝑐, (𝑎𝛼𝑏)𝛽𝑐=𝑎𝛼(𝑏𝛽𝑐).

For any 𝑎,𝑏∈𝑀 and for 𝛼,𝛽∈Γ the expressions 𝑎𝛼𝑏−𝑏𝛼𝑎 is denoted by [𝑎,𝑏]_𝛼 and 𝛼𝑎𝛽−𝛽𝑎𝛼 are denoted by [𝛼,𝛽]_𝑎. Then one has the following identities:𝑎𝛽𝑏,𝑐_𝛼=𝑎𝛽[𝑏,𝑐]_𝛼+[𝑎,𝑐]_𝛼𝛽𝑏+𝑎𝛽,𝛼_𝑐𝑏,𝑎,𝑏𝛽𝑐_𝛼=𝑏𝛽[𝑎,𝑐]_𝛼+[𝑎,𝑏]_𝛼𝛽𝑐+𝑏𝛽,𝛼_𝑎𝑐, (2.1) for all 𝑎,𝑏,𝑐∈𝑀 and for all 𝛼,𝛽∈Γ. Using the assumption (*) the above identities reduce to𝑎𝛽𝑏,𝑐_𝛼=𝑎𝛽[𝑏,𝑐]_𝛼+[𝑎,𝑐]_𝛼𝛽𝑏,𝑎,𝑏𝛽𝑐_𝛼=𝑏𝛽[𝑎,𝑐]_𝛼+[𝑎,𝑏]_𝛼𝛽𝑐, (2.2) for all 𝑎,𝑏,𝑐∈𝑀 and 𝛼, 𝛽∈Γ.

Further 𝑀 stands for a prime Γ-ring with center 𝑍(𝑀). The ring 𝑀 is 𝑛-torsion-free if 𝑛𝑥=0, 𝑥∈𝑀 implies 𝑥=0, where 𝑛 is a positive integer, 𝑀 is prime if 𝑎Γ𝑀Γ𝑏=0 implies 𝑎=0 or 𝑏=0, and it is semiprime if 𝑎Γ𝑀Γ𝑎=0 implies 𝑎=0. An additive mapping 𝑇∶𝑀→𝑀 is called a left (right) centralizer if 𝑇(𝑥𝛼𝑦)=𝑇(𝑥)𝛼𝑦(𝑇(𝑥𝛼𝑦)=𝑥𝛼𝑇(𝑦)) for 𝑥,𝑦∈𝑀, 𝛼∈Γ and it is called a Jordan left (right) centralizer if𝑇(𝑥𝛼𝑥)=𝑇(𝑥)𝛼𝑥(𝑇(𝑥𝛼𝑥)=𝑥𝛼𝑇(𝑥))∀𝑥∈𝑀,𝛼∈Γ. (2.3) A mapping 𝜃∶𝑀×𝑀→𝑀 is called biadditive if it is additive in both arguments. An additive mapping 𝐷∶𝑀→𝑀 is called a derivation if 𝐷(𝑥𝛼𝑦)=𝐷(𝑥)𝛼𝑦+𝑥𝛼𝐷(𝑦) for all 𝑥,𝑦∈𝑀, 𝛼∈Γ and it is called a Jordan derivation if 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥+𝑥𝛼𝐷(𝑥) for all 𝑥∈𝑀, 𝛼∈Γ. A derivation 𝐷 is inner if there exists 𝑎∈𝑀, such that 𝐷(𝑥)=𝑎𝛼𝑥−𝑥𝛼𝑎 holds for all 𝑥∈𝑀, 𝛼∈Γ. Every derivation is a Jordan derivation. The converse is in general not true. An additive mapping 𝐷∶𝑀→𝑀 is said to be a generalized derivation if there exists a derivation 𝑑∶𝑀→𝑀 such that 𝐷(𝑥𝛼𝑦)=𝐷(𝑥)𝛼𝑦+𝑥𝛼𝑑(𝑦) for all 𝑥,𝑦∈𝑀, 𝛼∈Γ. The maps of the form 𝑥→𝑎𝛼𝑥+𝑥𝛼𝑏 where 𝑎, 𝑏 are fixed elements in 𝑀 and for all 𝛼∈Γ called the generalized inner derivation. An additive mapping 𝐷∶𝑀→𝑀 is said to be a Jordan generalized derivation if there exists a derivation 𝑑∶𝑀→𝑀 such that 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥+𝑥𝛼𝑑(𝑥) for all 𝑥∈𝑀, 𝛼∈Γ. 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 [4].

3. Main Results

We start from the following subsidiary results.

Lemma 3.1. Let 𝑀 be a semiprime Γ-ring. If 𝑎,𝑏∈𝑀 are such that 𝑎𝛼𝑥𝛽𝑏=0 for all 𝑥∈𝑀, 𝛼,𝛽∈Γ, then 𝑎𝛼𝑏=𝑏𝛼𝑎=0.

Proof. Let 𝑥∈𝑀. Then 𝑎𝛼𝑏𝛽𝑥𝛾𝑎𝛼𝑏=𝑎𝛼𝑏𝛽𝑥𝛾𝑎𝛼𝑏=0,𝑏𝛼𝑎𝛽𝑥𝛾𝑏𝛼𝑎=𝑏𝛼𝑎𝛽𝑥𝛾𝑏𝛼𝑎=0. (3.1) By semiprimeness of 𝑀 with respect to 𝛽,𝛾∈Γ, it follows that 𝑎𝛼𝑏=𝑏𝛼𝑎=0.

Lemma 3.2. Let 𝑀 be a semiprime Γ-ring and 𝜃,𝜑∶𝑀×𝑀→𝑀 biadditive mappings. If 𝜃(𝑥,𝑦)𝛼𝑤𝛽𝜑(𝑥,𝑦)=0 for all 𝑥,𝑦,𝑤∈𝑀, then 𝜃(𝑥,𝑦)𝛼𝑤𝛽𝜑(𝑢,𝑣)=0 for all 𝑥,𝑦,𝑢,𝑣,𝑤∈𝑀, 𝛼,𝛽∈Γ.

Proof. First we replace 𝑥 with 𝑥+𝑢 in the relation 𝜃(𝑥,𝑦)𝛼𝑤𝛽𝜑(𝑥,𝑦)=0, and use the biadditivity of the 𝜃 and 𝜑. Then we have 𝜃𝑥+𝑢,𝑦𝛼𝑤𝛽𝜑𝑥+𝑢,𝑦=0⟹𝜃𝑥,𝑦𝛼𝑤𝛽𝜑𝑢,𝑦=−𝜃𝑢,𝑦𝛼𝑤𝛽𝜑𝑥,𝑦. (3.2) Then 𝜃𝜃𝜃𝜃𝑥,𝑦𝛼𝑤𝛽𝜑𝑢,𝑦𝛾𝑧𝛿𝑥,𝑦𝛼𝑤𝛽𝜑𝑢,𝑦=−𝑢,𝑦𝛼𝑤𝛽𝜑𝑥,𝑦𝛾𝑧𝛿𝑢,𝑦𝛼𝑤𝛽𝜑𝑢,𝑦=0. (3.3) Hence 𝜃(𝑥,𝑦)𝛼𝑤𝛽𝜑(𝑢,𝑦)=0 by semiprimeness of 𝑀 with respect to 𝛾,𝛿∈Γ.
Now we replace 𝑦 by 𝑦+𝑣 and obtain the assertion of the lemma with the similar observation as above.

Lemma 3.3. Let 𝑀 be a semiprime Γ-ring satisfying the assumption (*) and 𝑎 be a fixed element of 𝑀. If 𝑎𝛽[𝑥,𝑦]_𝛼=0 for all 𝑥,𝑦∈𝑀, 𝛼,𝛽∈Γ, then 𝑎∈𝑍(𝑀).

Proof. First we calculate the following expressions using the assumption (*), [𝑧,𝑎]_𝛼𝛽𝑥𝛿[𝑧,𝑎]_𝛼=𝑧𝛼𝑎𝛽𝑥𝛿[𝑧,𝑎]_𝛼−𝑎𝛼𝑧𝛽𝑥𝛿[𝑧,𝑎]_𝛼=𝑧𝛼𝑎𝛽[𝑧,𝑥𝛼𝑎]_𝛿−𝑧𝛼𝑎𝛽[𝑧,𝑥]_𝛿𝛼𝑎−𝑎𝛼[𝑧,𝑧𝛿𝑥𝛼𝑎]_𝛽+𝑎𝛼[𝑧,𝑧𝛿𝑥]_𝛽𝛼𝑎. (3.4) Since 𝑎𝛽[𝑥,𝑦]_𝛼=0 for all 𝑥,𝑦∈𝑀, 𝛼,𝛽∈Γ, we get [𝑧,𝑎]_𝛼𝛽𝑥𝛿[𝑧,𝑎]_𝛼=0. By the semiprimeness of 𝑀 we get [𝑧,𝑎]_𝛼=0 for all 𝛼∈Γ. Hence 𝑎∈𝑍(𝑀).

Lemma 3.4. Let 𝑀 be a Γ-ring satisfying the condition (*) and 𝐷∶𝑀→𝑀 be a Jordan generalized derivation with the associated derivation 𝑑. Let 𝑥,𝑦,𝑧∈𝑀 and 𝛼,𝛽∈Γ. Then(i)𝐷(𝑥𝛼𝑦+𝑦𝛼𝑥)=𝐷(𝑥)𝛼𝑦+𝐷(𝑦)𝛼𝑥+𝑥𝛼𝑑(𝑦)+𝑦𝛼𝑑(𝑥). In particular, if 𝑀 is 2-torsion-free, then(ii)𝐷(𝑥𝛼𝑦𝛽𝑥)=𝐷(𝑥)𝛼𝑦𝛽𝑥+𝑥𝛽𝑑(𝑦𝛽𝑥),(iii)𝐷(𝑥𝛼𝑦𝛽𝑧+𝑧𝛽𝑦𝛼𝑥)=𝐷(𝑥)𝛼𝑦𝛽𝑧+𝐷(𝑧)𝛽𝑦𝛼𝑥+𝑥𝛼𝑑(𝑦𝛽𝑧)+𝑧𝛽𝑑(𝑦𝛼𝑥).

Proof. (i) We have 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥+𝑥𝛼𝑑(𝑥), for all 𝑥∈𝑀, 𝛼∈Γ. Then replacing 𝑥 by 𝑥+𝑦, and following the series of implications below we get the result: 𝐷𝛼𝛼+(𝑥)𝑦(𝑥)𝑦𝑦(𝑥)𝑦(𝑥)=𝐷(𝑥)(𝑥)𝑦𝑦(𝑥)𝑦𝑦(𝑥)(𝑥)𝑦𝑦(𝑥)𝑥+𝑦𝑥+𝑦=𝐷𝑥+𝑦𝑥+𝑦𝑥+𝑦𝛼𝑑𝑥+𝑦⟹𝐷𝑥𝛼𝑥+𝑥𝛼𝑦+𝑦𝛼𝑥+𝑦𝛼𝑦=𝐷𝛼𝑥+𝐷𝛼𝑥+𝐷𝛼𝑦+𝐷𝛼𝑥+𝑥𝛼𝑑+𝑥𝛼𝑑+𝑦𝛼𝑑+𝑦𝛼𝑑⟹𝐷𝑥𝛼𝑥+𝑦𝛼𝑦+𝐷𝑥𝛼𝑦+𝑦𝛼𝑥𝛼𝑥+𝑥𝛼𝑑+𝐷𝛼𝑦+𝑦𝛼𝑑+𝐷𝛼𝑦+𝐷𝛼𝑥+𝑥𝛼𝑑+𝑦𝛼𝑑⟹𝐷𝑥𝛼𝑦+𝑦𝛼𝑥=𝐷𝛼𝑦+𝐷𝛼𝑥+𝑥𝛼𝑑+𝑦𝛼𝑑,∀𝑥,𝑦∈𝑀,𝛼∈Γ. (3.5)
(ii) Replace 𝑦 by 𝑥𝛽𝑦+𝑦𝛽𝑥 in the above relation (3.5), then we get,𝐷+(𝑥)𝛼+(𝑥)(𝑥)(𝑥)(𝑥)𝑦𝑦(𝑥)+(𝑥)𝑥𝛼𝑥𝛽𝑦+𝑦𝛽𝑥𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑥=𝐷𝑥𝛽𝑦+𝑦𝛽𝑥+𝐷𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑥+𝑥𝛼𝑑𝑥𝛽𝑦+𝑦𝛽𝑥𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑑,∀𝑥,𝑦,𝑧∈𝑀,𝛼,𝛽∈Γ.⟹𝐷𝑥𝛼𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑥+𝐷𝑥𝛼𝑦𝛽𝑥+𝐷𝑥𝛽𝑦𝛼𝑥=𝐷𝛼𝑥𝛽𝑦+𝐷𝛼𝑦𝛽𝑥+𝐷𝛽𝑦𝛼𝑥+𝐷𝛽𝑥𝛼𝑥+𝑥𝛽𝑑𝛼𝑥+𝑦𝛽𝑑𝛼𝑥+𝑥𝛼𝑑𝑥𝛽𝑦+𝑦𝛽𝑥𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑑,∀𝑥,𝑦,𝑧∈𝑀,𝛼,𝛽∈Γ. (3.6) Using the assumption (*), we conclude that 𝐷(𝑥)(𝑥)(𝑥)𝑦𝑦(𝑥)+(𝑥)𝑥𝛼𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑥+2𝐷𝑥𝛼𝑦𝛽𝑥=𝐷𝛼𝑥𝛽𝑦+𝐷𝛼𝑦𝛽𝑥+𝐷𝛽𝑦𝛼𝑥+𝐷𝛽𝑥𝛼𝑥+𝑥𝛽𝑑𝛼𝑥+𝑦𝛽𝑑𝛼𝑥+𝑥𝛼𝑑𝑥𝛽𝑦+𝑦𝛽𝑥𝑥𝛽𝑦+𝑦𝛽𝑥𝛼𝑑,∀𝑥,𝑦,𝑧∈𝑀,𝛼,𝛽∈Γ. (3.7) Again, replacing 𝑥 by 𝑥𝛽𝑥 in (3.5) 𝐷𝑦𝑦(𝑥)(𝑥)𝑦𝑦(𝑥)(𝑥)𝑥𝛽𝑥𝛼𝑦+𝑦𝛼𝑥𝛽𝑥=𝐷𝑥𝛽𝑥𝛼𝑦+𝐷𝛼𝑥𝛽𝑥+𝑥𝛽𝑥𝛼𝑑+𝑦𝛼𝑑𝑥𝛽𝑥,∀𝑥,𝑦∈𝑀,𝛼,𝛽∈Γ,⟹𝐷𝑥𝛽𝑥𝛼𝑦+𝑦𝛼𝑥𝛽𝑥=𝐷𝛽𝑥𝛼𝑦+𝑥𝛽𝑑𝛼𝑦+𝐷𝛼𝑥𝛽𝑥+𝑥𝛽𝑥𝛼𝑑+𝑦𝛼𝑑𝛽𝑥+𝑦𝛼𝑥𝛽𝑑,∀𝑥,𝑦∈𝑀,𝛼,𝛽∈Γ. (3.8) Adding both sides 2𝐷(𝑥𝛼𝑦𝛽𝑥), we get, (𝑥)(𝑥)𝑦𝑦(𝑥)(𝑥)⟹𝐷𝑥𝛽𝑥𝛼𝑦+𝑦𝛼𝑥𝛽𝑥+2𝐷𝑥𝛼𝑦𝛽𝑥=𝐷𝛽𝑥𝛼𝑦+𝑥𝛽𝑑𝛼𝑦+𝐷𝛼𝑥𝛽𝑥+𝑥𝛽𝑥𝛼𝑑+𝑦𝛼𝑑𝛽𝑥+𝑦𝛼𝑥𝛽𝑑+2𝐷𝑥𝛼𝑦𝛽𝑥,∀𝑥,𝑦∈𝑀,𝛼,𝛽∈Γ. (3.9) Comparing (3.7) and (3.9) we obtain, 2𝐷𝑥𝛼𝑦𝛽𝑥=2𝐷(𝑥)𝛼𝑦𝛽𝑥+𝑥𝛼𝑦𝛽𝑑(𝑥)+𝑥𝛼𝑑𝑦𝛽𝑥. (3.10) Since 𝑀 is 2-torsion-free, it gives 𝐷𝑥𝛼𝑦𝛽𝑥=𝐷(𝑥)𝛼𝑦𝛽𝑥+𝑥𝛼𝑑𝑦𝛽𝑥. (3.11)
(iii) Replace 𝑥 for 𝑥+𝑧 in (3.12), we get,𝐷()()()()+()()(𝑥)(𝑧)(𝑥)(𝑧)=𝐷(𝑥)(𝑧)+𝐷(𝑧)(𝑥)(𝑥)(𝑧)(𝑥)(𝑧)𝑥+𝑧𝛼𝑦𝛽𝑥+𝑧=𝐷𝑥+𝑧𝛼𝑦𝛽𝑥+𝑧𝑥+𝑧𝛼𝑑𝑦𝛽𝑥+𝑧⟹𝐷𝑥𝛼𝑦𝛽𝑥+𝑥𝛼𝑦𝛽𝑧+𝑧𝛼𝑦𝛽𝑥+𝑧𝛼𝑦𝛽𝑧=𝐷𝛼𝑦𝛽𝑥+𝐷𝛼𝑦𝛽𝑥+𝐷𝛼𝑦𝛽𝑧+𝐷𝛼𝑦𝛽𝑧+𝑥𝛼𝑑𝑦𝛽𝑥+𝑥𝛼𝑑𝑦𝛽𝑧+𝑧𝛼𝑑𝑦𝛽𝑥+𝑦𝛽𝑧⟹𝐷𝑥𝛼𝑦𝛽𝑥+𝑧𝛼𝑦𝛽𝑧+𝐷𝑥𝛼𝑦𝛽𝑧+𝑧𝛼𝑦𝛽𝑥𝛼𝑦𝛽𝑥+𝐷𝛼𝑦𝛽𝑧+𝑥𝛼𝑑𝑦𝛽𝑥+𝑧𝛼𝑑𝑦𝛽𝑧𝛼𝑦𝛽𝑥+𝐷𝛼𝑦𝛽𝑧+𝑥𝛼𝑑𝑦𝛽𝑧+𝑧𝛼𝑑𝑦𝛽𝑥⟹𝐷𝑥𝛼𝑦𝛽𝑧+𝑧𝛼𝑦𝛽𝑥=𝐷𝛼𝑦𝛽𝑧+𝐷𝛼𝑦𝛽𝑥+𝑥𝛼𝑑𝑦𝛽𝑧+𝑧𝛼𝑑𝑦𝛽𝑥⟹𝐷𝑥𝛼𝑦𝛽𝑧+𝑧𝛽𝑦𝛼𝑥=𝐷𝛼𝑦𝛽𝑧+𝐷𝛼𝑦𝛽𝑥+𝑥𝛼𝑑𝑦𝛽𝑧+𝑧𝛼𝑑𝑦𝛽𝑥,∀𝑥,𝑦∈𝑀,𝛼,𝛽∈Γ. (3.12)

Definition 3.5. Let 𝑀 be a Γ-ring and 𝐷∶𝑀→𝑀 be a Jordan generalized derivation with the associated derivation 𝑑. Define 𝐺_𝛼(𝑥,𝑦)=𝐷(𝑥𝛼𝑦)−𝐷(𝑥)𝛼𝑦−𝑥𝛼𝑑(𝑦) for all 𝑥,𝑦∈𝑀 and 𝛼∈Γ.

Lemma 3.6. The function 𝐺_𝛼(𝑥,𝑦) has the following properties:(i)𝐺_𝛼(𝑥,𝑦)+𝐺_𝛼(𝑦,𝑥)=0. (ii)𝐺_𝛼(𝑥,𝑦+𝑧)=𝐺_𝛼(𝑥,𝑦)+𝐺_𝛼(𝑥,𝑧). (iii)𝐺_𝛼(𝑥+𝑦,𝑧)=𝐺_𝛼(𝑥,𝑧)+𝐺_𝛼(𝑦,𝑧). (iv)𝐺_𝛼+𝛽(𝑥,𝑦)=𝐺_𝛼(𝑥,𝑦)+𝐺_𝛽(𝑥,𝑦).

Proof. The results easily follow from Lemma 3.4(i).

Remark 3.7. 𝐷 is a generalized derivation if and only if 𝐺_𝛼(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑀, 𝛼∈Γ.

Theorem 3.8. Let 𝑀 be a 2-torsion-free semiprime Γ-ring satisfying the condition (*). Let 𝑥,𝑦,𝑧∈𝑀 and 𝛽,𝛿∈Γ. Then(i)𝐺_𝛼(𝑥,𝑦)𝛽𝑧𝛿[𝑥,𝑦]_𝛼=0 for all 𝑧∈𝑀,(ii)𝐺_𝛼(𝑥,𝑦)𝛽[𝑥,𝑦]_𝛼=0.

Proof. (i) From Lemma 3.4(iii) we get 𝐷𝑥𝛼𝑦𝛽𝑧+𝑧𝛽𝑦𝛼𝑥=𝐷(𝑥)𝛼𝑦𝛽𝑧+𝐷(𝑧)𝛽𝑦𝛼𝑥+𝑥𝛼𝑑𝑦𝛽𝑧+𝑧𝛽𝑑𝑦𝛼𝑥. (3.13) We set 𝐴=𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦. (3.14) Since 𝐷(𝐴)(𝑥)𝑦(𝑥)𝑦𝑦(𝑥)=𝐷𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦=𝐷𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦=𝐷𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑥𝛼𝑑𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝐷𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦+𝑦𝛼𝑑𝑥𝛽𝑧𝛿𝑥𝛼𝑦=𝐷𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑥𝛼𝑑𝛽𝑧𝛿𝑦𝛼𝑥+𝑥𝛼𝑦𝛽𝑑𝑧𝛿𝑦𝛼𝑥+𝐷𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦+𝑦𝛼𝑑𝛽𝑧𝛿𝑥𝛼𝑦+𝑦𝛼𝑥𝛽𝑑𝑧𝛿𝑥𝛼𝑦,∀𝑥,𝑦,𝑧∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.15) Again 𝐷(𝐴)+=𝐷𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦=𝐷𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦=𝐷𝑥𝛼𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝐷𝑦𝛼𝑥𝛽𝑧𝛿𝑥𝛼𝑦+𝑥𝛼𝑦𝛽𝑑𝑧𝛿𝑦𝛼𝑥+𝑦𝛼𝑥𝛽𝑑𝑧𝛿𝑥𝛼𝑦,∀𝑥,𝑦,𝑧∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.16) From (3.15) and (3.16) we find, 𝐷𝑥𝛼𝑦−𝐷(𝑥)𝛼𝑦−𝑥𝛼𝑑𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝐷𝑦𝛼𝑥−𝐷𝑦𝛼𝑥−𝑦𝛼𝑑(𝑥)𝛽𝑧𝛿𝑥𝛼𝑦=0.⟹𝐺_𝛼𝑥,𝑦𝛽𝑧𝛿𝑦𝛼𝑥+𝐺_𝛼𝑦,𝑥𝛽𝑧𝛿𝑥𝛼𝑦=0,⟹𝐺_𝛼𝑥,𝑦𝛽𝑧𝛿𝑦𝛼𝑥−𝐺_𝛼𝑥,𝑦𝛽𝑧𝛿𝑥𝛼𝑦=0,⟹𝐺_𝛼𝑥,𝑦𝛽𝑧𝛿𝑦,𝑥_𝛼=0,⟹𝐺_𝛼𝑥,𝑦𝛽𝑧𝛿𝑥,𝑦_𝛼=0,for𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.17)
(ii) According to Lemma 3.4(ii) we have,𝐷𝑥𝛼𝑦𝛽𝑧=𝐷(𝑥)𝛼𝑦𝛽𝑧+𝑥𝛼𝑑𝑦𝛽𝑧. (3.18) Replace 𝑧 by 𝑥𝛼𝑦 in (3.18) we find 𝐷(𝑥)𝛽(𝑥)(𝑥)⟹𝐷(𝑥)𝑦𝑥𝛼𝑦𝛽𝑥𝛼𝑦=𝐷𝛼𝑦𝛽𝑥𝛼𝑦+𝑥𝛼𝑑𝑦𝛽𝑥𝛼𝑦⟹𝐷𝑥𝛼𝑦𝑥𝛼𝑦=𝐷𝛼𝑦𝛽𝑥𝛼𝑦+𝑥𝛼𝑑𝑦𝛽𝑥𝛼𝑦⟹𝐷𝑥𝛼𝑦𝛽𝑥𝛼𝑦+𝑥𝛼𝑦𝛽𝑑𝑥𝛼𝑦−𝐷𝛼𝑦𝛽𝑥𝛼𝑦−𝑥𝛼𝑑𝑦𝛽𝑥𝛼𝑦=0𝑥𝛼𝑦−𝐷𝛼𝑦−𝑥𝛼𝑑𝛽𝑥𝛼𝑦+𝑥𝛼𝑦𝛽𝑑𝑥𝛼𝑦−𝑥𝛼𝑦𝛽𝑑𝑥𝛼𝑦=0⟹𝐺_𝛼𝑥,𝑦𝛽𝑥𝛼𝑦=0. (3.19) Similarly 𝐺_𝛼(𝑥,𝑦)𝛽𝑦𝛼𝑥=0. Therefore, 𝐺_𝛼𝑥,𝑦𝛽𝑥𝛼𝑦−𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥=𝐺_𝛼𝑥,𝑦𝛽𝑥,𝑦_𝛼=0. (3.20)

Lemma 3.9. 𝐺_𝛼(𝑥,𝑦)∈𝑍(𝑀), for all 𝑥,𝑦∈𝑀, 𝛼∈Γ.

Proof. From Theorem 3.8(ii), we have 𝐺_𝛼(𝑥,𝑦)𝛽[𝑥,𝑦]_𝛼=0. Therefore due to Lemma 3.4(ii) 𝐺_𝛼(𝑥,𝑦)∈𝑍(𝑀).

Theorem 3.10. Let 𝑀 be a 2-torsion-free semiprime Γ-ring satisfying the assumption (*) and 𝐷∶𝑀→𝑀 be a Jordan generalized derivation with associated derivation 𝑑 on 𝑀. Then 𝐷 is a generalized derivation.

Proof. In particular, 𝑟𝛾𝐺_𝛼(𝑥,𝑦), 𝐺_𝛼(𝑥,𝑦)𝛾𝑟∈𝑍(𝑀) for all 𝑟∈𝑀, 𝛼,𝛾∈Γ. This gives 𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦=𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥=𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥=𝑦𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑥. (3.21) Then 4𝐷(𝑥𝛼𝐺_𝛼(𝑥,𝑦)𝛿𝐺_𝛼(𝑥,𝑦)𝛽𝑦)=4𝐷(𝑦𝛽𝐺_𝛼(𝑥,𝑦)𝛿𝐺_𝛼(𝑥,𝑦)𝛼𝑥). Now we will compute each side of this equality by using (3.15) and the above relation, 4𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦=2𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼(𝑥)𝑥,𝑦𝛽𝑦𝛼𝑥=2𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝑥𝛼𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦+2𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼(𝑥)(𝑥)𝑥,𝑦𝛽𝑦𝛼𝑑=2𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+2𝑥𝛼𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼(𝑥)(𝑥)𝑥,𝑦𝛽𝑦𝛼𝑑=2𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑑𝑦𝛼𝑥+𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+2𝑥𝛼𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑑(𝑥). (3.22) So we get 4𝐷𝑥𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼(𝑥)𝑥,𝑦𝛼𝑦=2𝐷𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑦+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦𝑥+𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑑𝑦𝛼𝑥+𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+2𝑥𝛼𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑑(𝑥),𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.23) Moreover, 4𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥=2𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑦𝑥,𝑦𝛽𝑥𝛼𝑦=2𝐷𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+2𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑥𝛼𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑦𝑦𝑥,𝑦𝛽𝑥𝛼𝑑=2𝐷𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑥+𝑥𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑦+2𝑦𝛽𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑦𝑦𝑥,𝑦𝛽𝑥𝛼𝑑=2𝐷𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑥𝛼𝑦+𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑥𝛼𝑦+𝐷(𝑥)𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑦+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑑(𝑥)𝛽𝑦+𝑥𝛼𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦+2𝑦𝛽𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑑𝑦. (3.24) So we get 4𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑦𝑥,𝑦𝛼𝑥=2𝐷𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦+𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦+𝐷(𝑥)𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑦+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑑(𝑥)𝛽𝑦+𝑥𝛼𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦+2𝑦𝛽𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑑𝑦,𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.25) Comparing the results of (3.22) and (3.24) and using the above relations 𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥=𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥=𝑥𝛼𝐺_𝛼𝑥,𝑦𝛽𝑦=𝑥𝛼𝐺_𝛼𝑥,𝑦𝛽𝑦=𝐺_𝛼𝐺𝑥,𝑦𝛽𝑥𝛼𝑦,_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥=𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦𝛼𝑥=𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦=𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦,𝑥𝛽𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑦=𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦=𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑥𝛼𝑦=𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑦𝛽𝑥=𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥=𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥, (3.26) we obtain 𝐷(𝑥)𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑦𝑦𝑥,𝑦𝛽𝑑=𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑥+𝑦𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑑(𝑥), (3.27) which gives 𝜑_𝛼𝑥,𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦=𝜑_𝛼𝑦,𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦, (3.28) where 𝜑_𝛼(𝑥,𝑦) stands for 𝐷(𝑥)𝛼𝑦+𝑥𝛼𝑑(𝑦).
On the other hand, we have4𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦=4𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝑦𝛽𝐺_𝛼𝑥,𝑦. (3.29) Now we use (3.15) and the properties of 𝐺_𝛼(𝑥,𝑦), to derive 4𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦=2𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦=2𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦+2𝑥𝛼𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦+2𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑑𝑥𝛼𝑦, (3.30) which gives 4𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦=2𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦+2𝑥𝛼𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦+2𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑑𝑥𝛼𝑦,𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.31) Moreover, 4𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝑦𝛽𝐺_𝛼𝑥,𝑦=2𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝑦𝛽𝐺_𝛼𝑥,𝑦+𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝑥𝛼𝐺_𝛼𝐺𝑥,𝑦=2𝐷_𝛼𝑥,𝑦𝛼𝑥𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝑑𝐺_𝛼𝐺𝑥,𝑦𝛽𝑦+2𝐷_𝛼𝑥,𝑦𝛽𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥=𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝑑𝐺_𝛼(𝑥)𝑥,𝑦𝛼𝑥=𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥𝛽𝑦+𝑥𝛼𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐺_𝛼𝑥,𝑦𝛼𝑑(𝑥)𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛽𝑑𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥. (3.32) So we obtain 4𝐷𝑥𝛼𝐺_𝛼𝑥,𝑦𝛿𝑦𝛽𝐺_𝛼(𝑥)𝑥,𝑦=𝐷𝛼𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛽𝑥𝛼𝑦+𝑥𝛼𝑑_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐺_𝛼𝑥,𝑦𝛼𝑑(𝑥)𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦+2𝐺_𝛼𝑥,𝑦𝛼𝑥𝛿𝑑𝐺_𝛼𝑥,𝑦𝛽𝑦+𝐷𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝐺𝑥,𝑦𝛼𝑥+𝐷_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦𝛼𝑥+𝑦𝛽𝑑𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+𝐺_𝛼𝑥,𝑦𝛽𝑑𝑦𝛿𝐺_𝛼𝑥,𝑦𝛼𝑥+2𝐺_𝛼𝑥,𝑦𝛽𝑦𝛿𝑑𝐺_𝛼𝑥,𝑦𝛼𝑥,𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.33) Comparing (3.31) and (3.33), we derive 2𝐷𝑥𝛼𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦=𝜑_𝛼𝑥,𝑦𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦+𝜑_𝛼𝑦,𝑥𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦,𝑥,𝑦∈𝑀,𝛼,𝛽,𝛿∈Γ. (3.34) Finally using (3.31) we get 𝐷(𝑥𝛼𝑦)𝛽𝐺_𝛼(𝑥,𝑦)𝛿𝐺_𝛼(𝑥,𝑦)=𝜑_𝛼(𝑥,𝑦)𝛽𝐺_𝛼(𝑥,𝑦)𝛿𝐺_𝛼(𝑥,𝑦). But 𝐺_𝛼(𝑥,𝑦)=𝐷(𝑥𝛼𝑦)−𝜑_𝛼(𝑥,𝑦). By the semiprimeness of 𝑀, we have 𝜑_𝛼(𝑥,𝑦)𝛽𝐺_𝛼(𝑥,𝑦)=0. Again by the primeness of 𝑀, we get 𝐺_𝛼(𝑥,𝑦). The proof is complete.

It is clear that if we let the derivation 𝑑 to be the zero derivation in the above theorem, we get the following result.

Theorem 3.11. Let 𝑀 be a 2-torsion-free semiprime Γ-ring and 𝐷∶𝑀→𝑀 be an additive mapping which satisfies 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥 for all 𝑥∈𝑀, 𝛼∈Γ. Then 𝐷 is a left centralizer

Proof. We have 𝐷(𝑥𝛼𝑥)=𝐷(𝑥)𝛼𝑥. (3.35) If we replace 𝑥 by 𝑥+𝑦, we get 𝐷𝑥𝛼𝑦+𝑦𝛼𝑥=𝐷(𝑥)𝛼𝑦+𝐷𝑦𝛼𝑥. (3.36) By replacing 𝑦 with 𝑥𝛼𝑦+𝑦𝛼𝑥 and using (3.5), we arrive at 𝐷𝑥𝛼𝑥𝛼𝑦+𝑦𝛼𝑥+𝑥𝛼𝑦+𝑦𝛼𝑥𝛼𝑥=𝐷(𝑥)𝛼𝑥𝛼𝑦+𝐷(𝑥)𝛼𝑦𝛼𝑥+𝐷(𝑥)𝛼𝑦𝛼𝑥+𝐷𝑦𝛼𝑥𝛼𝑥. (3.37) But on the other hand, 𝐷𝑥𝛼𝑥𝛼𝑦+𝑦𝛼𝑥𝛼𝑥+2𝐷𝑥𝛼𝑦𝛼𝑥=𝐷(𝑥)𝛼𝑥𝛼𝑦+𝐷𝑦𝛼𝑥𝛼𝑥+2𝐷𝑥𝛼𝑦𝛼𝑥. (3.38) Comparing (3.37) and (3.38) we obtain 𝐷𝑥𝛼𝑦𝛼𝑥=𝐷(𝑥)𝛼𝑦𝛼𝑥. (3.39) If we linearize (3.39) in 𝑥, we get 𝐷𝑥𝛼𝑦𝛼𝑧+𝑧𝛼𝑦𝛼𝑥=𝐷(𝑥)𝛼𝑦𝛼𝑧+𝐷(𝑧)𝛼𝑦𝛼𝑥. (3.40) Now we shall compute 𝐷(𝑥𝛼𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝑦𝛼𝑥𝛼𝑧𝛼𝑥𝛼𝑦) in two different ways. If we use (3.39) we have 𝐷𝑥𝛼𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝑦𝛼𝑥𝛼𝑧𝛼𝑥𝛼𝑦=𝐷(𝑥)𝛼𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝐷𝑦𝛼𝑥𝛼𝑧𝛼𝑥𝛼𝑦. (3.41) But if we use (3.40) we have 𝐷𝑥𝛼𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝑦𝛼𝑥𝛼𝑧𝛼𝑥𝛼𝑦=𝐷𝑥𝛼𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝐷𝑦𝛼𝑥𝛼𝑧𝛼𝑥𝛼𝑦. (3.42) Comparing (3.41) and (3.42) and introducing a bi-additive mapping 𝐺_𝛼(𝑥,𝑦)=𝐷(𝑥𝛼𝑦)−𝐷(𝑥)𝛼𝑦 we arrive at 𝐺_𝛼𝑥,𝑦𝛼𝑧𝛼𝑦𝛼𝑥+𝐺_𝛼𝑦,𝑥𝛼𝑧𝛼𝑥𝛼𝑦=0. (3.43) Equality (3.36) can be rewritten in this notation as 𝐺_𝛼(𝑥,𝑦)=−𝐺_𝛼(𝑦,𝑥). Using this fact and (3.43) we obtain 𝐺_𝛼𝑥,𝑦𝛼𝑧𝛼𝑥,𝑦_𝛼=0. (3.44) Using first Lemma 3.2 and then Lemma 3.1 we have 𝐺_𝛼𝑥,𝑦𝛼𝑧𝛼[𝑢,𝑣]_𝛼=0. (3.45) Now fix some 𝑥,𝑦∈𝑀 and using Lemma 3.3 we get 𝐺_𝛼(𝑥,𝑦)∈𝑍.
In particular, 𝑟𝛽𝐺_𝛼(𝑥,𝑦), 𝐺_𝛼(𝑥,𝑦)𝛽𝑟∈𝑍 for all 𝑟∈𝑀. This gives𝑥𝛽𝐺_𝛼𝑥,𝑦𝛿𝐺_𝛼𝑥,𝑦𝛽𝑦=𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝑥=𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥=𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥. (3.46) Therefore 4𝐷(𝑥Γ𝐺_𝛼(𝑥,𝑦)Γ𝐺_𝛼(𝑥,𝑦)Γ𝑦)=4𝐷(𝑦Γ𝐺_𝛼(𝑥,𝑦)Γ𝐺_𝛼(𝑥,𝑦)Γ𝑥). Both sides of this equality will be computed in few steps using (3.36), 2𝐷𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝑥=2𝐷𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼,(𝑥)𝑥,𝑦Γ𝑥Γ𝑦2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+2𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑦𝑥,𝑦Γ𝑦Γ𝑥=2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+2𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼(𝑥)𝑥,𝑦Γ𝑥Γ𝑦,2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑦𝑥,𝑦Γ𝑥=2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼(𝑥)𝑥,𝑦Γ𝑦,2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝑥+𝐷𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑦𝑥,𝑦Γ𝑥=2𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼(𝑥)𝑥,𝑦Γ𝑥Γ𝑦+𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝐷(𝑥)𝑥,𝑦Γ𝑦,Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑦𝑥,𝑦Γ𝑦Γ𝑥=𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝑦. (3.47) Since 𝐺_𝛼(𝑥,𝑦)Γ𝑦Γ𝑥=𝐺_𝛼(𝑥,𝑦)Γ𝑦Γ𝑥=𝑥Γ𝐺_𝛼(𝑥,𝑦)Γ𝑦=𝑥Γ𝐺_𝛼(𝑥,𝑦)Γ𝑦=𝐺_𝛼(𝑥,𝑦)Γ𝑥Γ𝑦, we obtain 𝐷(𝑥)Γ𝐺_𝛼𝑥,𝑦Γ𝑦=𝐷𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥. (3.48) On the other hand, we also have 4𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦=4𝐷𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝐺_𝛼𝑥,𝑦,2𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝑦=2𝐷𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝐺_𝛼𝑥,𝑦+𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝐺_𝛼𝑥,𝑦,2𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦+2𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝐺𝑥,𝑦Γ𝑥Γ𝑦=2𝐷_𝛼𝑥,𝑦Γ𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝑦+2𝐷𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥,2𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦+2𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝑦=𝐷𝑥Γ𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐷𝑦Γ𝐺_𝛼𝑥,𝑦+𝐺_𝛼𝑥,𝑦Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥,2𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦+2𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼(𝑥)𝑥,𝑦Γ𝑥Γ𝑦=𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑦+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑦𝑥,𝑦Γ𝑥Γ𝑦+𝐷Γ𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥+𝐷𝐺_𝛼𝑥,𝑦Γ𝐺_𝛼𝑥,𝑦Γ𝑥Γ𝑦,2𝐷𝑥Γ𝑦Γ𝐺_𝛼𝑥