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On semi-transitional and transitional rings

Sourav Koner, Titas Saha, Biswajit Mitra

TL;DR

The paper develops a unified algebra–topology framework by introducing semi-transitional rings and transitional rings, organized around semi-transition maps $\psi$ that encode zero-set behavior under ring operations. It establishes localization and product preservation, analyzes ideals via $\varphi$-filters and $\varphi$-ultrafilters, and characterizes semiprimitive and pm-rings within this setting. A key contribution is the Omega-compactification $\Omega_f(\Lambda)$, a Stone–Čech–like compactification of the maximal spectrum with a universal extension property for maps into compact spaces, yielding deep connections to classical compactifications such as $\beta N$ and $\beta X$. The framework encompasses classical examples like $k[x]$, $C^*(X)$, and $I$-convergent sequence rings $A_I$, unifying several known constructions under a common algebraic-topological lens and enabling robust unique-extension results in topological algebra.

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.

On semi-transitional and transitional rings

TL;DR

The paper develops a unified algebra–topology framework by introducing semi-transitional rings and transitional rings, organized around semi-transition maps that encode zero-set behavior under ring operations. It establishes localization and product preservation, analyzes ideals via -filters and -ultrafilters, and characterizes semiprimitive and pm-rings within this setting. A key contribution is the Omega-compactification , a Stone–Čech–like compactification of the maximal spectrum with a universal extension property for maps into compact spaces, yielding deep connections to classical compactifications such as and . The framework encompasses classical examples like , , and -convergent sequence rings , unifying several known constructions under a common algebraic-topological lens and enabling robust unique-extension results in topological algebra.

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.

Paper Structure

This paper contains 8 sections, 23 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Semi-transitional rings
  3. Ideals and induced filters associated with semi-transition maps
  4. Structural representation of semiprimitive associated semi-transitional rings
  5. Transitional rings
  6. An important class of transitional rings
  7. Omega-compactification
  8. Application

Key Result

Theorem 2.4

Let $F$ be a nonempty index set and let $\psi_{\alpha}: A_{\alpha} \rightarrow \raisebox{.15\baselineskip}{\Large$\wp$}(\Lambda_{\alpha})$ be a semi-transition map for each $\alpha \in F$. Suppose there exists a fixed positive integer $m$ such that for every $\alpha \in F$, $\psi_{\alpha}(a_{\alpha}

Theorems & Definitions (55)

  • Definition 2.1
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Example 2.8
  • Example 2.9
  • Theorem 3.1
  • ...and 45 more