Metrically complete regular rings

Abstract:

This paper is concerned with the structure of those (von Neumann) regular rings $R$ which are complete with respect to the weakest metric derived from the pseudo-rank functions on $R$, known as the ${N^ \ast }$-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all ${\aleph _0}$-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group ${K_0}(R)$, which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of $R$. For instance, it is proved that the simple homomorphic images of $R$ are right and left self-injective rings, and $R$ is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective $R$-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of $R$. As another example of the results derived, it is proved that if all simple artinian homomorphic images of $R$ are $n \times n$ matrix rings (for some fixed positive integer $n$), then $R$ is an $n \times n$ matrix ring.