Commutativity of Two Torsion Free σ-Prime Gamma Rings with Nonzero Derivations

Abstract

Let M be a 2-torsion free σ-prime Γ-ring and d a nonzero derivation on M. Then M is commutative with the help of the condition [d(x),x]α∈Z(M), for all x∈M and α∈Γ. Let I be a nonzero σ-ideal of M and d a nonzero derivation on M commuting with σ. Then M is commutative in both conditions [d(x),d(y)]α=0 and d(xαy) = d(yαx), for all x,y∈I and α∈Γ.