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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the structure of those (von Neumann) regular rings which are complete with respect to the weakest metric derived from the pseudo-rank functions on , known as the -metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all -continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group , 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 . For instance, it is proved that the simple homomorphic images of are right and left self-injective rings, and is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective -modules are determined by the isomorphism classes modulo the maximal two-sided ideals of . As another example of the results derived, it is proved that if all simple artinian homomorphic images of are matrix rings (for some fixed positive integer ), then is an matrix ring.

**[1]**Erik M. Alfsen,*Compact convex sets and boundary integrals*, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. MR**0445271****[2]**K. R. Goodearl,*von Neumann regular rings*, Monographs and Studies in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR**533669****[3]**K. R. Goodearl and D. E. Handelman,*Metric completions of partially ordered abelian groups*, Indiana Univ. Math. J.**29**(1980), no. 6, 861–895. MR**589651**, 10.1512/iumj.1980.29.29060**[4]**K. R. Goodearl, D. E. Handelman, and J. W. Lawrence,*Affine representations of Grothendieck groups and applications to Rickart 𝐶*-algebras and ℵ₀-continuous regular rings*, Mem. Amer. Math. Soc.**26**(1980), no. 234, vii+163. MR**571998**, 10.1090/memo/0234**[5]**David Handelman,*Representing rank complete continuous rings*, Canad. J. Math.**28**(1976), no. 6, 1320–1331. MR**0432694****[6]**David Handelman,*Simple regular rings with a unique rank function*, J. Algebra**42**(1976), no. 1, 60–80. MR**0417233****[7]**-, Finite Rickart*-algebras and their properties*, Studies in Analysis, Advances in Math. Suppl. Studies, Vol. 4, 1979, pp. 171-196.**[8]**David Handelman, Denis Higgs, and John Lawrence,*Directed abelian groups, countably continuous rings, and Rickart 𝐶*-algebras*, J. London Math. Soc. (2)**21**(1980), no. 2, 193–202. MR**575375**, 10.1112/jlms/s2-21.2.193**[9]**G. L. Seever,*Measures on 𝐹-spaces*, Trans. Amer. Math. Soc.**133**(1968), 267–280. MR**0226386**, 10.1090/S0002-9947-1968-0226386-5**[10]**John von Neumann,*Continuous geometry*, Foreword by Israel Halperin. Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. MR**0120174**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
16A30,
46A55,
46L99

Retrieve articles in all journals with MSC: 16A30, 46A55, 46L99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0656490-6

Keywords:
Regular ring,
-complete,
,
interpolation group

Article copyright:
© Copyright 1982
American Mathematical Society