International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 157169, 8 pages
http://dx.doi.org/10.1155/2012/157169
Research Article

Noetherian and Artinian Lattices

Derya Keskin Tütüncü,1 Sultan Eylem Toksoy,2 and Rachid Tribak3

1Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey
2Department of Mathematics, İzmir Institute of Technology, 35430 Urla, İzmir, Turkey
3Centre Pédagogique Régional (CPR) Tanger, Avenue My Abdelaziz Souani, BP 3117, Tangier 90000, Morocco

Received 30 March 2012; Accepted 12 April 2012

Academic Editor: Palle E. Jorgensen

Copyright © 2012 Derya Keskin Tütüncü et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is proved that if L is a complete modular lattice which is compactly generated, then Rad(L)/0 is Artinian if, and only if for every small element a of L, the sublattice a/0 is Artinian if, and only if L satisfies DCC on small elements.