On commutativity of completely prime gamma-rings

• Main Authors: Rakhimov, Isamiddin Sattarovich, Dey, Kalyan Kumar, Paul, Akhil Chandra
• Format: Article
• Language: English
• Published: Universiti Putra Malaysia Press 2013
• Online Access: http://psasir.upm.edu.my/id/eprint/30157/1/30157.pdf; http://psasir.upm.edu.my/id/eprint/30157/; http://einspem.upm.edu.my/journal/volume7.2.php

In this paper we prove that any completely prime Γ -ring M satisfying the condition aαbβc=aβbαc = ( a,b,c ∈ M and α, β∈Γ )with nonzero derivation, is a commutative integral Γ -domain if its characteristic is not two. We also show that if the characteristic of M is 2 the Γ -ring M is either commutative or is an order in a simple 4-dimensional algebra over its center. We give necessary condition in terms of derivations for belongings of an element of the Γ -ring M to the center of M when the characteristic of M is not two. If char M = 2, and a Z M ∉ ( ), then we show that the derivation is the inner derivation.