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

Abstract

We find a distributive (v, 0, 1)-semilattice S 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. - There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to S. These results are established by constructing an infinitary statement, denoted here by URPsr, 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