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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distributive lattices with countably many compact elements ([Von Neumann regular rings with tailor-made ideal lattices, unpublished notes (1986)]; several proofs are given in a paper of the author and the reviewer [Algebra Univers. 45, No. 1, 71-102 (2001; Zbl 1039.06003)]), but the author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
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