# zbMATH — the first resource for mathematics

Representation of algebraic distributive lattices with $$\aleph_1$$ compact elements as ideal lattices of regular rings. (English) Zbl 0989.16010
A well known, easy fact is that any lattice appearing as the lattice of ideals of a (von Neumann) regular ring is algebraic and distributive. G. M. Bergman proved the converse for algebraic distributive lattices with countably many compact elements ([Von Neumann regular rings with tailor-made ideal lattices, unpublished notes (1986)]; several proofs are given in a paper of the author and the reviewer [Algebra Univers. 45, No. 1, 71-102 (2001; Zbl 1039.06003)]), but the author showed that the converse can fail when $$\aleph_2$$ compact elements are present [Proc. Am. Math. Soc. 127, No. 2, 363-370 (1999; Zbl 0902.06006)]. Here he fills the gap: Any algebraic distributive lattice $$D$$ with at most $$\aleph_1$$ compact elements is isomorphic to the ideal lattice of some regular ring $$R$$, and if the largest element of $$D$$ is compact, then $$R$$ is unital. Unlike Bergman’s construction, the author’s method (based on an amalgamation theorem for rings and semilattices which is interesting in its own right) produces regular rings that are not unit-regular, nor even locally matricial; in fact, they satisfy the property that every finitely generated projective module is isomorphic to its double ($$A\cong A\oplus A$$). It remains an open problem whether an algebraic distributive lattice with at most $$\aleph_1$$ compact elements must appear as the ideal lattice of some locally matricial regular ring.
As a corollary to the main theorem, the author obtains that any algebraic distributive lattice $$D$$ with at most $$\aleph_1$$ compact elements is isomorphic to the congruence lattice of some sectionally complemented modular lattice $$L$$. This improves on results of A. P. Huhn [Acta Sci. Math. 53, No. 1-2, 3-10, 11-18 (1989; Zbl 0684.06009, Zbl 0684.06010)], who obtained such a representation with $$L$$ only a lattice, and G. Grätzer, H. Lakser and the author [Acta Sci. Math. 66, No. 1-2, 3-22 (2000; Zbl 0953.06010)], who were able to do it with $$L$$ sectionally complemented.

##### MSC:
 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 06A12 Semilattices 16D25 Ideals in associative algebras 06C20 Complemented modular lattices, continuous geometries 06B15 Representation theory of lattices
Full Text: