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

Abstract

Abstract. We find a distributive (∨, 0, 1)-semilattice Sω 1 of size ℵ 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 Sω 1 . -There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to Sω 1 . 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 Sω 1 .

AF-embeddings into C∗-algebras of real rank zero

“…The preceding corollary was already known (although maybe not with this precise formulation) and is related to a result of G. M. Bergman about realizing distributive algebraic lattices as ideal lattices of von Neumann regular rings ( [2], see also [28], [15], [29]). We make the connection more explicit as follows (see also the comments after Theorem 4.5).…”

The Realization Problem for Von Neumann Regular Rings

“…If L is an algebraic distributive lattice which is not the congruence lattice of any lattice and M is any conical refinement monoid such that Id(M ) ∼ = L, then M cannot be realized as V(R) for a regular ring R. For every algebraic distributive lattice L there is at least one such conical refinement monoid, namely the semilattice L c of compact elements of L, we should expect a myriad of such monoids to exist. Wehrung proved in [39] that if |L c | ≤ ℵ 1 then the semilattice L c can be realized as V(R) where R is a von Neumann regular ring, and he showed in [40] that there is a distributive semilattice S ω1 of size ℵ 1 which is not the semilattice of finitely generated, idempotent-generated ideals of any exchange ring of finite stable rank. In particular there is no locally matricial K-algebra A over a field K such that Id c (A) ∼ = S ω1 ; see [40] for details.…”

“…Wehrung proved in [39] that if |L c | ≤ ℵ 1 then the semilattice L c can be realized as V(R) where R is a von Neumann regular ring, and he showed in [40] that there is a distributive semilattice S ω1 of size ℵ 1 which is not the semilattice of finitely generated, idempotent-generated ideals of any exchange ring of finite stable rank. In particular there is no locally matricial K-algebra A over a field K such that Id c (A) ∼ = S ω1 ; see [40] for details. This contrasts with Bergman's result [13] stating that every distributive semilattice of size ≤ ℵ 0 is the semilattice of finitely generated ideals of an ultramatricial K-algebra, for every field K.…”

A solution to Dilworth's congruence lattice problem

“…A new uniform refinement property. In many works such as [25,29,32,33,36,39,40], the classes of semilattices that are representable with respect to various functors are separated from the corresponding counterexamples by infinitary statements called uniform refinement properties. We shall now discuss briefly how this can also be done here.…”

“…However, the latter result has been extended further by Růžička [27], who proved that the representing lattice can be taken relatively complemented, modular, and locally finite. This is not possible for (1) above, as, for |S| ≤ ℵ 1 , one can take L relatively complemented modular [37], relatively complemented and locally finite [12], but not necessarily both [39].…”