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.
Categories: math.RA