Modules whose submodule lattice is lower finite

Abstract

A bounded poset [Formula: see text] is said to be lower finite if [Formula: see text] is infinite and for all [Formula: see text], there are but finitely many [Formula: see text] such that [Formula: see text] In this paper, we classify the modules [Formula: see text] over a commutative ring [Formula: see text] with identity with the property that the lattice [Formula: see text] of [Formula: see text]-submodules [Formula: see text] (under set-theoretic containment) is lower finite. Our results are summarized in Theorem 3.1 at the end of this note.