A Right Radical for Right D.G. Near-Rings

Abstract

Rahbari embarked on developing a right representation theory for right d.g. near-rings and proved interesting right structure theorems. Our main focus is connections between left and right representation. We discuss a Jacobson-type radical, ^rJ₀(R), for right d.g. near-rings. The radical ^rJ₀(R) is defined using annihilators of certain d.g. right R-groups which are the equivalent of type-0 R-groups from left representation. We then explore connections in near-rings with suitable chain conditions between ^rJ₀(R), the (left) radicals and the intersection of all maximal right ideals, denoted ^rJ_1/2(R). In particular we prove that J₂(R) = ^rJ₀(R) for near-rings R satisfying the descending chain condition for left R-subgroups of R⁺.