On the Pure-injectivity Profile of a Ring

Abstract

An analog of the injective profile of a ring, with relative injectivity replaced by relative pure-injectivity, is investigated. Emphasis is placed on comparing and contrasting the properties of both profiles. In particular, the analog in this context of the notion of poor modules is considered and properties of pure-injectively poor modules are determined. While we do not know of any ring that does not have pure-injectively poor modules, their existence has not been determined in general. Rings having pure-injectively poor modules of various types are characterized.

Modules with minimal copure-injectivity domain

“…For a right R-module M the class PI −1 (M ) = {N ∈ Mod-R | M is N -pure-injective} is said to be the pure-injectivity domain of M . In [8], pureinjectively poor (shortly pi-poor ) modules were introduced as modules with minimal pure-injectivity domain and studied. Although the existence of pure-injectively poor abelian groups is proved in [2], we do not know whether pi-poor modules exist over arbitrary rings (see [8]).…”

“…Rings over which every right R-module is copi-poor are shown to be right CDS rings. In [8], it is proved that R is a right PDS ring if and only if every right R-module is pi-poor. Since commutative PDS rings are CDS (see [12]), a copi-poor module need not be pi-poor in general and conversely.…”

Poor and pi-poor Abelian groups

“…In [7], the authors investigate the notion of pi-poor module and study properties of these modules over various rings. In particular they give some classes of groups that are not pi-poor and point out that they do not know whether a pi-poor group exists or not.…”

“…It is not known whether pi-poor modules exists over arbitrary rings. In particular in [7] some classes of abelian groups that are not pi-poor are given but the authors point out that they do not know whether a pi-poor abelian group exists.…”

“…Let K and N be right R-modules. K is N -pure-injective if for each pure submodule L of N every homomorphism f : L → K can be extended to a homomorphism g : N → K. Following [7], a right R-module M is called pureinjectively poor (or simply pi-poor) if whenever M is N -pure-injective, then N is pure-split. It is not known whether pi-poor modules exists over arbitrary rings.…”