Semilattices of finitely generated ideals of exchange rings with finite stable rank
Friedrich Wehrung · Transactions of the American Mathematical Society · 2003
We find a distributive ( ∨ , 0 , 1 ) (\vee ,0,1) -semilattice S ω 1 S_{\omega _1} of size ℵ 1 \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 ω 1 S_{\omega _1} . [—] There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to S ω 1 S_{\omega _1} . These results are established by constructing an infinitary statement, denoted here by U R P s r \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 ω 1 S_{\omega _1} .