Algebraic Behavior of Idempotent Elements in Near Ring

Abstract

The study of Boolean near-rings and idempotent elements forms an important part of modern algebra, particularly in exploring the structural behavior of near-rings and their algebraic properties. In this work, we investigate the nature of idempotent elements in near-rings with special emphasis on near-fields and Boolean near-rings. We show that:

1.Near fields have exactly two idempotent elements. We claim that they are the trivial idempotent elements. Further, we give concrete example to support this claim that if R is not a near field, the idempotent elements are not exactly two.

2.Idempotent element is a near integral domain also trivial.

3.Every Boolean near ring is commutative near ring.

Overall, our findings provide a deeper understanding of the interplay between idempotent elements, near-fields, near integral domains, and Boolean near-rings. The results not only contribute to the theoretical advancement of near-ring theory but also establish useful connections with commutativity, algebraic simplicity, and structural classification. This study opens pathways for further exploration of near-rings in relation to other algebraic systems and their applications in both pure and applied mathematics.

Moreover, future research can extend these results to hybrid algebraic systems, examine categorical frameworks that unify near-rings with related algebraic structures, and explore their interdisciplinary use in computer science, data encryption, and logical reasoning models, thereby strengthening both the theoretical and applied dimensions of near-ring theory.