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Regularity and $\mathsf{K}_0$-Regularity under Finiteness Conditions

Rafael Parra

Abstract

The purpose of this work is to investigate various notions of regularity from the perspective of finiteness conditions, with the ultimate goal of identifying broad classes of rings that are $\mathsf{K}_0$-regular. In this direction, we revisit the classical concepts of coherence and von Neumann regularity, and establish new characterizations. We then focus on the study of \emph{$n$-coherent regular rings}, recently introduced in [31], and analyze their $\mathsf{K}$-theoretic behavior. Finally, we present applications illustrating how these approaches provide examples of $\mathsf{K}_0$-regular rings.

Regularity and $\mathsf{K}_0$-Regularity under Finiteness Conditions

Abstract

The purpose of this work is to investigate various notions of regularity from the perspective of finiteness conditions, with the ultimate goal of identifying broad classes of rings that are -regular. In this direction, we revisit the classical concepts of coherence and von Neumann regularity, and establish new characterizations. We then focus on the study of \emph{-coherent regular rings}, recently introduced in [31], and analyze their -theoretic behavior. Finally, we present applications illustrating how these approaches provide examples of -regular rings.

Paper Structure

This paper contains 20 sections, 94 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cotorsion pair.
  4. Pseudo-Coherence, Coherence, and von Neumann Regularity
  5. Von Neumann regularity
  6. Regularity and Regular Coherence
  7. Commuatative case
  8. Semi-Regularity
  9. Stably Coherent Rings
  10. $n$-Coherence and Regularity
  11. Commutative case
  12. Vogel-Regularity
  13. Uniform $n$-regularity
  14. Commutative case
  15. Finiteness Conditions and $\mathsf{K}_0$-Regularity
  16. ...and 5 more sections

Key Result

Proposition 3.1

Let $R$ be a ring. The class of pseudo-coherent $R$-modules is closed under submodules, extensions and direct sums. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (177)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Corollary 3.6
  • Proposition 3.7
  • ...and 167 more