An alternative perspective on flatness of modules

Abstract

Given modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be absolutely [Formula: see text]-pure if [Formula: see text] is a monomorphism for every extension [Formula: see text] of [Formula: see text]. For a module [Formula: see text], the absolutely pure domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is absolutely [Formula: see text]-pure. As an opposite to flatness, a module [Formula: see text] is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and off-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.