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JMB Journal of Microbiolog and Biotechnology

OPEN ACCESS eISSN 0454-8124
pISSN 1225-6951
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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

Abstract

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