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Semilattices of finitely generated ideals of exchange rings with finite stable rank. (English) Zbl 1034.06007
The author gives an example of a distributive $$(\vee, 0,1)$$-semilattice $$S_{w_1}$$ 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 (the origin of this work is in the representation problem of distributive $$(\vee,0)$$-semilattices as semilattices of compact ideals of von Neumann regular rings). So the author obtains solutions to various open problems in ring theory and lattice theory, such as:
– 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 $$S_{w_1}$$.
– There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $$S_{w_1}$$.
These results are established by constructing an infinitary statement 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 $$S_{w_1}$$.

##### MSC:
 06A12 Semilattices 20M14 Commutative semigroups 06B10 Lattice ideals, congruence relations 19K14 $$K_0$$ as an ordered group, traces
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