A radical for near-rings

Abstract

SynopsisThroughout this paper the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. That is, x · 0 = 0 and (x + y)z = xz + yz for all x, y, z ∈ N. In what follows we generalise the notion of s-primitivity first introduced in an earlier paper by the author (1968), where only distributively generated (d.g.) near-rings with identity were considered. We define a Jacobson type radical Js (N) and show that J1(N)⊇Js(N) ⊇ Q(N), where Q(N) is the intersection of all 0-modular left ideals of N (Pilz). In addition we settle some of the problems remaining from Hartney (1968).