Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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  $S_{\omega_1}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 06A12, 20M14, 06B10, 19K14

Retrieve articles in all journals with MSC (2000): 06A12, 20M14, 06B10, 19K14


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: http://dx.doi.org/10.1090/S0002-9947-03-03369-5
PII: S 0002-9947(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