On quasi-regularity in gamma near-rings

Abstract

The Jacobson radicals, $$J_\nu $$ , ( $$\nu = 0, 1, 2$$ ), of $$\Gamma $$ -near-rings were introduced and studied by Booth. In this paper quasi-regular elements in a $$\Gamma $$ -near-ring are introduced and a characterization of the $$J_0$$ -radical of a $$\Gamma $$ -near-ring in terms of quasi-regular ideals is given. It is also proved that $$J_0(M)$$ is nilpotent for a $$\Gamma $$ -near-ring M with DCC on $$M\Gamma $$ -subgroups of M. It is verified that if M is a $$\Gamma $$ -near-ring satisfying DCC on $$M \Gamma $$ -subgroups of M then $$J_{2}(M) = J_{1}(M) = J_{1/2}(M) = J_{0}(M)$$ .