Polynomial near-rings, distributor and J2 ideals of generalized centralizer near-rings

Abstract

This work has three main parts. In the first section we define and develop some structure theory for a new near-ring which, for special oases, coincides with well-known polynomial ring theory results. In particular, we develop a set of tools for investigating the near-ring and further consider in part the questions of ideal structures, the J2 -radical ideal, the distributor ideal and the corresponding quotient near-ring structures. This new near-ring is by definition different from the class of com position polynomial near-lings. In the second part, we determine the J2-radieal and the distributor ideal of generalized centralizer near-rings. Some comparisons of the two are also included. The final section presents a counter-example to the question "Does R prime imply that M[R[(R^2) is simple?", a question raised in the work of Maxson and van der W alt in [6].