On ( σ , τ )-Derivations and Commutativity of Prime and Semi prime Γ-rings

Let R be a Г-ring, and σ, τ be two automorphisms of R. An additive mapping d from a Γ-ring R into itself is called a (σ,τ)-derivation on R if d(aαb) = d(a)α σ(b) + τ(a)αd(b), holds for all a,b ∈R and α∈Γ. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)]α = [ ] ( ) holds for all a,b∈R and α∈Γ. In this paper, we investigate the commutativity of R by the strong commutativity preserving (σ,τ)derivation d satisfied some properties, when R is prime and semi prime Г-ring.


Introduction
Let R and Γ be two additive abelian groups.If for any a, b, c ∈R and α,β ∈Γ, the following conditions are satisfied, (i) a α b ∈R (ii) (a+b)αc = aαc + bαc, a(α +β)b =aαb + aβb, aα(b + c) = aαb + aαc (iii) (aαb)βc = aα(bβc), then R is called a Γ-ring (see [4]).The set Z(R) = {a ∈R| aαb = bαa, ∀b ∈R, andα∈Γ} is called the center of R.A Γ-ring R is called prime if aΓRΓb = 0 with a, b ∈R implies a = 0 or b = 0, and R is called semi prime if aΓRΓa = 0 with a ∈R implies a = 0.The notion of a (resp.semi-) prime Γ-ring is an extension for the notion of a (resp.semi-) prime ring.In [1] holds for all a, b ∈R and α∈Γ.The notion of a strong commutativity preserving map was first introduced by Bell and Mason [3], and in [4] X. Jing Ma, and Y. Zhou proved that a semi prime Γ-ring with a strong commutativity preserving derivation on itself must be commutative.In this paper, we obtain that a Γ-ring R with a strong Open Access commutativity preserving (σ,τ)derivation d on itself must be commutative, when R is prime and semi prime Г-ring.We write [a, b] α = aαb -bαa.Throughout this paper R will denote a Γ-ring satisfying an assumption (*)….aαbβc = aβbαc, for all a,b,c ∈R and α,β∈ Γ .We will often use the identities:

Main Results
First we prove the following lemmas.

Lemma 2:
Let R be 2-torsion free prime and d be a(σ,τ)-derivation, and d can be commuted with σ and τ.If d 2 = 0 then d = 0.

Proof:
Let, for all a, b∈R, α∈Γ.From the hypothesis we have

Theorem 3:
Let R be a prime Γ-ring with a non zero (σ,τ)-derivation d.If d is a strong commutativity preserving then R is commutative ring.

Proof:
For all a,b∈R and α∈Γ, we have By using lemma 1 in (11), we get [r, t] α =0 , for all r ,t ∈R and α ∈Γ.Hence R is commutative ring.

Theorem 4:
Let R be a prime Γ-ring, and d is a strong commutativity preserving (σ,τ)derivation.If σ = τ then either R is commutative ring or d =0.

Proof:
By hypothesis, we have =0 , for all c∈R and α∈Γ Replacing τ(c) by c and b by bλd(c) in (4) and using (1),we have F.J.Jing defined a derivation on Γ-ring as follows, an additive map d from a Γring R into itself is called a derivation on R if d(aαb) = d(a)αb +aαd(b) , holds for all a,b∈R and α∈Γ, and in [2] S. Ali and M.Salahudin Khan defined (σ,τ)-derivation on R ,for two endomorphism σ and τ as follows: an additive map d from R into R is called a (σ,τ)-derivation on R if d(aαb) = d(a)ασ(b) + τ(a)αd(b) , holds for all a,b ∈R and α ∈Γ.