On generalized derivations and commutativity of prime gamma-rings

Abstract

Let M be a prime Γ-ring with center Z(M), I a nonzero ideal of M and F be a generalized derivation with associated nonzero derivation d. In the present paper, our purpose is to produce commutativity results for prime Γ-rings M admitting a generalized derivation F satisfying any one of the properties: (i) F(xαy)∓xαy ∈ Z(M), (ii) F(xαy)∓yαx ∈ Z(M), (iii) F(x)αF(y)∓xαy ∈ Z(M), (iv) F(x)αF(y)∓yαx ∈ Z(M), (v) F([x, y]_α)=[F(x), y]_α, (vi) [F(x), y]_α∓[x, F(y)]_α=0, (vii) F([x, y]_α)∓[d(x), d(y)]_α=0, for all x,y ∈ I and α ∈ Γ. Also, some examples are given to show that the primeness of the various results is not superfluous.