Dual Krull dimension and quotient finite dimensionality

Abstract

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x, 1] has no infinite independent set for any x epsilon L. We characterize upper continuous modular lattices L that have dual Krull dimension k(0) (L) less than or equal to alpha, by relating that with the property of L being QFD and with other conditions involving subdirectly irreducible lattices and/or meet irreducible elements. In particular, we answer in the positive, in the more general latticial setting, some open questions on QFD modules raised by Albu and Rizvi [Comm. Algebra 29 (2001) 1909-1928]. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.