Riv.Mat.Univ.Parma (5) 2 (1993)-Parte I


On near-rings in which the ideals are annihilators

Pages 1-10
Received: 28 July 1992  
AMS Classification: 16Y30

Introduction Rings R in which all (left or right) ideals are (left or right) annihilators of subsets of R are studied in [4], [5], [6], [8]. The purpose of this paper is to extend the above situation to near-rings and to establish the structure theory for R-near-rings: namely the near-rings N in which every non-trivial ideal is a right annihilaator of a subset of N.

Using [2], we characterize the R-near rings which contain an ideal I such that its left annihilator is without nonzero nilpotent elements, and we prove, in particular, that such near-rings are subdirectly reducible and thus they have the right annihilator that equals {0}.
Moreover, we show that, by adding some little stronger conditions, we find integral near-rings in which all ideals are prime, linearly ordered and with integral factors. Thus such near-rings result special cases of near-rings studied in [3].