Lattices of two-sided ideals of locally matricial algebras and the \(\Gamma\)-invariant problem
It is well known that the lattice of two-sided ideals of a von Neumann regular ring is distributive. G. M. Bergman proved that every algebraic distributive lattice is either isomorphic to the lattice of lower subsets of a partially ordered set or, with at most countably many compact elements, is isomorphic to the two-sided ideal lattice of a locally matricial algebra. In contrast, \textit{F. Wehrung} [Isr. J. Math. 103, 177--206 (1998; Zbl 0916.06018)] constructed an algebraic distributive lattice with \(\aleph_2\) compact elements which cannot be realized as the lattice of two-sided ideals of any von Neumann regular ring. Further, he proved that if an algebraic distributive lattice has \(\aleph_1\) compact elements, then it can be realized as the lattice of two-sided ideals of a von Neumannr regular ring; however he proved recently that the result fails for locally matricial algebras. The main result of the present paper is the representation of every algebraic distributive lattice whose compact elements form a lattice as the lattice of a locally matricial algebra.
scientific article; zbMATH DE number 2108111