On Commutativity of σ-Prime Γ-Rings
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Let U be a σ-square closed Lie ideal of a 2-torsion free σ-prime Γ-ring M. Let d≠1 be an automorphism of M such that [u, d(u)]α ∈ Z(M) on U, dσ=σd on U, and there exists u0 in Saσ(M) with MΓu0 ⊆ U. Then, U ⊆ Z(M). By applying this result, we generalize the results of Oukhtite and Salhi respect to Γ-rings. Finally, for a non-zero derivation of a 2-torsion free σ-prime Γ-ring, we obtain suitable conditions under which the Γ-ring must be commutative.
Abstract
Let U be a σ-square closed Lie ideal of a 2-torsion free σ-prime Γ-ring M. Let d≠1 be an automorphism of M such that [u, d(u)]α ∈ Z(M) on U, dσ=σd on U, and there exists u0 in Saσ(M) with MΓu0 ⊆ U. Then, U ⊆ Z(M). By applying this result, we generalize the results of Oukhtite and Salhi respect to Γ-rings. Finally, for a non-zero derivation of a 2-torsion free σ-prime Γ-ring, we obtain suitable conditions under which the Γ-ring must be commutative.
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