A Study on the Classification of All Simple Commutative Near-Rings upto Isomorphism

Abstract

This research paper aims to classify all simple commutative Near-Rings up to isomorphism, which are rings with no zero divisors and no ideals containing a nonzero element that is not prime ideal. The authors use various techniques from algebraic number theory and representation theory of semi simple algebras to construct representations of these rings over the integers or rational numbers. They also provide examples showing how these classes can be distinguished by their algebraic properties, such as whether they are PID (prime idempotent domain), zero-divisorless, or have ideals containing elements with prime index. The study contributes new results and insights into the classification of near- rings, which has implications for other areas of abstract algebra and number theory.