Regularity and K0-Regularity Under Finiteness Conditions

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 Formula: see text-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 Formula: see text-coherent regular rings, recently introduced in 32, and analyze their Formula: see text-theoretic behavior. Finally, we present applications illustrating how these approaches provide examples of Formula: see text-regular rings.