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 ∈ L . We characterize upper continuous modular lattices L that have dual Krull dimension k 0 ( L ) ⩽ α , 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.