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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Metrically complete regular rings

Author: K. R. Goodearl
Journal: Trans. Amer. Math. Soc. 272 (1982), 275-310
MSC: Primary 16A30; Secondary 46A55, 46L99
MathSciNet review: 656490
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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.

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Keywords: Regular ring, $ {N^ \ast }$-complete, $ {K_0}$, interpolation group
Article copyright: © Copyright 1982 American Mathematical Society