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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
Published electronically: October 28, 2003
MathSciNet review: 2031048
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Abstract: We find a distributive \ensuremath{(\vee,0,1)}-semilattice $S_{\omega_1}$ of size $\aleph_1$ 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:

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  $S_{\omega_1}$.

There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $S_{\omega_1}$.
These results are established by constructing an infinitary statement, denoted here by $\mathrm{URP_{sr}}$, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice  $S_{\omega_1}$.

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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

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

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