Representation of algebraic distributive lattices with N1 compact elements as ideal lattices of regular rings

We prove the following result: Theorem. Every algebraic distributive lattice D with at most N-1 compact elements is isomorphic to the ideal lattice of a con on Neumann regular ring R. (By earlier results of the author the N-1 bound is optimal.) Therefore D is also isomorphic to the congruence lattice of a yet sectionally complemented modular lattice L. namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one call 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. Gratzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our results is an amalgamation theorem for semilattices and algebras (over division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma. and G. Gratzer, H. Lakser, and the author.