Original Article
Kyungpook Mathematical Journal 2015; 55(4): 827-835
Published online December 23, 2015
Copyright © Kyungpook Mathematical Journal.
On Commutativity of $sigma$-Prime $Gamma$-Rings
Kalyan Kumar Dey1 , Akhil Chandra Paul2,Bijan Davvaz3
Let $U$ be a $sigma$-square closed Lie ideal of a 2-torsion free $sigma$-prime $Gamma$-ring $M$. Let $d
ot =1$ be an automorphism of $M$ such that $[u, d(u)]_{alpha}in Z(M)$ on $U$, $dsigma =sigma d$ on $U$, and there exists $u_0$ in $Sa_{sigma} (M)$ with $MGamma u_0subseteq U$. Then, $U subseteq Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to $Gamma$-rings. Finally, for a non-zero derivation of a 2-torsion free $sigma$-prime $Gamma$-ring, we obtain suitable conditions under which the $Gamma$-ring must be commutative.
Keywords: $Gamma$-Rings with involution, $sigma$-prime $Gamma$-rings, centralizing automorphisms, square closed Lie ideals, derivations, , commutativity, $sigma$-square closed ideal