Abstract
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 = [a,b]α(σ,τ) 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.
Highlights
Let R and Γ be two additive abelian groups
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
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
Summary
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), R is called a Γ-ring (see [4]). 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. [1] 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
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