On semi-transitional and transitional rings

TL;DR

This paper introduces semi-transitional and transitional rings, extending classical ideas from rings of continuous functions, and explores their algebraic and topological properties, including ideal structure and compactification.

Contribution

It defines new classes of rings with a unified framework, generalizing known examples and analyzing their ideal and topological structures.

Findings

Characterization of prime and maximal ideals via semi transition maps

Conditions for semi transitional rings to be semiprimitive

Construction of a Stone Cech-like compactification for transitional rings

Abstract

In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general framework for these rings is developed through the notion of semi transition and transition maps, leading to a systematic exploration of their algebraic and topological properties. Structural results concerning product rings, localizations, and pm rings are established, showing that these new classes naturally generalize familiar examples such as polynomial rings over fields, rings of bounded continuous functions, and the ring of admissible ideal convergent real sequences. Ideals and filters induced by semi transition maps are analyzed to characterize prime and maximal ideals, revealing a duality between algebraic and set-theoretic constructions. Furthermore, conditions under which semi transitional rings become semiprimitive are determined, and a Stone Cech like compactification is constructed for transitional rings, giving rise to a new perspective on unique extension properties in topological algebra.