$\mathbf{k}_0$ of semiartinian von Neumann regular rings. Direct finiteness versus unit-regularity

TL;DR

This paper characterizes semiartinian von Neumann regular rings where unit-regularity, direct finiteness of factor rings, and the structure of the $K_0$ group are equivalent, providing a classification of their $K_0$ groups.

Contribution

It establishes the equivalence of key properties in semiartinian von Neumann regular rings and classifies the realizable $K_0$ groups for these rings.

Findings

Equivalence of unit-regularity and direct finiteness of factor rings.

$K_0(R)$ is free with a basis corresponding to simple modules.

Classification of realizable $K_0$ groups for semiartinian, unit-regular rings.

Abstract

If $R$ is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) $R$ is unit-regular, (2) every factor ring of $R$ is directly finite, (3) the abelian group $K_0(R)$ is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right $R$-modules. For the class of semiartinian and unit-regular rings the canonical partial order of $K_0(R)$ is investigated and the directed abelian groups which are realizable as $K_0(R)$ of these rings are classified.