We prove the following result:
Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.
(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively, that L has a largest element. This extends several known results of G. M. Bergman, A. P. Huhn, J. Tuma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10].
The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma, and G. Grätzer, H. Lakser, and the author.
@article{urn:eudml:doc:41405,
title = {Representation of algebraic distributive lattices with 1 compact elements as ideal lattices of regular rings.},
journal = {Publicacions Matem\`atiques},
volume = {44},
year = {2000},
pages = {419-435},
mrnumber = {MR1800815},
zbl = {0989.16010},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41405}
}
Wehrung, Friedrich. Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.. Publicacions Matemàtiques, Tome 44 (2000) pp. 419-435. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41405/