Semilattices of finitely generated ideals of exchange rings with finite stable rank

Author:
F. Wehrung

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1957-1970

MSC (2000):
Primary 06A12, 20M14, 06B10; Secondary 19K14

DOI:
https://doi.org/10.1090/S0002-9947-03-03369-5

Published electronically:
October 28, 2003

MathSciNet review:
2031048

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We find a distributive -semilattice of size that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular:

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- There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to .
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- There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to .

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Additional Information

**F. Wehrung**

Affiliation:
Département de Mathématiques, CNRS, UMR 6139, Université de Caen, Campus II, B.P. 5186, 14032 Caen Cedex, France

Email:
wehrung@math.unicaen.fr

DOI:
https://doi.org/10.1090/S0002-9947-03-03369-5

Keywords:
Semilattice,
distributive,
monoid,
refinement,
ideal,
stable rank,
strongly separative,
exchange ring,
lattice,
congruence

Received by editor(s):
January 3, 2003

Received by editor(s) in revised form:
April 2, 2003

Published electronically:
October 28, 2003

Article copyright:
© Copyright 2003
American Mathematical Society