Homological characterizations of almost Dedekind domains

Abstract

Let [Formula: see text] be a commutative ring. In this paper, a class of almost projective modules is introduced. An [Formula: see text]-module [Formula: see text] is said to be almost projective if [Formula: see text] for any [Formula: see text]-module [Formula: see text], where [Formula: see text] is a maximal ideal of [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] satisfying that [Formula: see text] is free over [Formula: see text] for any maximal ideal [Formula: see text] of [Formula: see text] is exactly almost projective. As applications, we characterize almost Dedekind domains in terms of almost projectivity and certain divisibility, respectively.