^{1}

^{*}

^{1}

^{2}

Let
M be a 2-torsion free semiprime
G
-ring with involution satisfying the condition that ( and ). In this paper, we will prove that if a non-zero Jordan
G
^{*}
-derivation d on M satisfies for all and , then .

The notion of G-ring was introduced as a generalized extension of the concept on classical ring. From its first appearance, the extensions and the generalizations of various important results in the theory of classical rings to the theory of G-rings have attracted a wider attention as an emerging field of research to enrich the world of algebra. A good number of prominent mathematicians have worked on this interesting area of research to develop many basic characterizations of G-rings. Nobusawa [

Let M and G be additive abelian groups. If there exists an additive mapping

1)

2)

3)

for all

A.......

According to assumption (A), the above commutator identities reduce to

During the past few decades, many authors have studied derivations in the context of prime and semiprime rings and G-rings with involution [

Definition 1 [

Definition 2. An element x in a G-ring M with involution is said to be hermitian if

Example 1. Let F be a field, and

Definition 3. An additive mapping

To further clarify the idea of

Example 2. Let R be a commutative ring with characteristic of R equal 2. Define

Define a mapping

To show that d is a

then

Now,

since

Definition 4. An additive mapping

Every

Example 3. Let M be a G-ring with involution and let

Define a mapping

for all

for all

then after reduction we get that d is a Jordan

for all

for all

then after reduction we get that d is not a

In this paper we will prove that if a non-zero Jordan

To prove our main results we need the following lemmas.

Lemma 1. Let M be a 2-torsion free semiprime G-ring with involution and

Proof. We have

for all

for all

for all

for all

for all

for all

for all

for all

for all

for all

for all

for all

hence by using assumption (A), we obtain

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for all

for all

for all

for all

for all

for all

for all

Lemma 2 Let M be a 2-torsion free semiprime G-ring with involution and

Proof. Putting

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for all

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for all

for all

Remark 1 [

Remark 2. Let M be a 2-torsion free simple G-ring with involution, then every

Proof. Define

Now

hence

and

hence

Therefore

hence

Theorem 1. Let M be a 2-torsion free semiprime G-ring with involution and

Proof. Assume that

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for all

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for all

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for all

for all

for all

Now assume

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This work is supported by the School of Mathematical Sciences, Universiti Sains Malaysia under the Short-term Grant 304/PMATHS/6313171.

Ali Kareem,Hajar Sulaiman,Abdul-Rahman Hameed Majeed, (2016) Jordan Γ*-Derivation on Semiprime Γ-Ring M with Involution. Advances in Linear Algebra & Matrix Theory,06,40-50. doi: 10.4236/alamt.2016.62006