Representation of algebraic distributive lattices with À1 compact elements as ideal lattices of regular rings.

Título inglés Representation of algebraic distributive lattices with À1 compact elements as ideal lattices of regular rings.
Título español Representación de retículos distributivos algebraicos con À1 elementos compactos como retículos de ideales de anillos regulares.
Autor/es Wehrung, Friedrich
Organización Dép. Math. Univ. Caen II, Caen, Francia
Revista 0214-1493
Publicación 2000, 44 (2): 419-435, 21 Ref.
Tipo de documento articulo
Idioma Inglés
Resumen inglés We prove the following result:

Theorem. Every algebraic distributive lattice D with at most À1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.

(By earlier results of the author, the À1 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.
Clasificación UNESCO 120108
Palabras clave español Retículos ; Anillos reticulados ; Anillos regulares
Código MathReviews MR1800815
Icono pdf Acceso al artículo completo