Published Paper Details:

This paper presents a detailed study on the behaviour of idempotent elements in the structural framework of stable and pseudo-stable near-rings. Idempotent elements play a central role in the decomposition and organization of algebraic systems, and their influence on the formation of ideals provides valuable insight into the internal properties of near-rings. The study of stability conditions within algebraic systems has been a central theme in understanding their internal symmetries and operational consistency. In this paper, we investigate stable and pseudo-stable properties in the context of near ring substructures. A near ring, being a generalization of a ring without the requirement of full distributivity or commutativity, provides a rich framework to analyze weaker forms of algebraic regularity. We define and explore the conditions under which substructures such as ideals, subnear-rings, and quasi-ideals exhibit stability under homomorphic images, endomorphisms, and composition mappings. The concept of pseudo-stability is introduced as a relaxed form of stability, wherein closure properties hold under specific mappings or restricted operations rather than universally. Comparative results between stable and pseudo-stable substructures are established, highlighting their interrelationships and dependencies on the structural constraints of the parent near ring. Illustrative examples and counterexamples are provided to demonstrate distinct behaviours within different classes of near rings. These findings contribute to the deeper understanding of near ring theory and open pathways for further exploration in algebraic systems exhibiting partial distributivity.