Subprojectivity domains of pure-projective modules

Abstract

As an alternative perspective on the injectivity of a pure-injective module, a pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module M is called subinjective relative to a module N if for every extension K of N , every homomorphism N → M can be extended to a homomorphism K → M . The subinjectivity domain of the module M is defined to be the class of modules N such that M is N -subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigent modules are studied. The structure of a ring over which every pure-injective (resp. simple, uniform, indecomposable) module is injective or subinjective relative only to the smallest possible family of modules is investigated. This work is a natural continuation to recent papers that have embraced the systematic study of the subinjective and subprojective domains of modules.

Subprojectivity in abelian categories

“…Similarly, in Proposition 3.7, we generalize [15,Proposition 2.14] by showing that the subprojectivity domain of a class L is closed under arbitrary direct products if and only if the subprojectivity domain of any of its objects is closed under arbitrary direct products. This result allows us to give a much direct proof (see Corollary 3.8) of a characterization of coherent rings established by Durgun in [9,Proposition 2.3]. Inspired by the work of Parra and Rada [18], we show that, if we assume further conditions on A , then the closure under direct products of the subprojectivity domains of classes can be characterized in terms of preenvelopes (see Proposition 3.11).…”

A Hyperstructural Approach to Semisimplicity

“…Moreover, in recent years, types of injectivity and projectivity have been studied with the help of (semi)simple modules. Refer to the books [10,11,[23][24][25] and the papers [26][27][28][29][30][31][32][33][34] for detailed information.…”

RD-projective module whose subprojectivity domain is minimal

“…We start by recalling what is understood by the notion of subprojectivity. Given modules X 1 and X 2 , X 1 is X 2 -subprojective if for each epimorphism α : P → X 2 and each morphism h : X 1 → X 2 , there exists a morphism f : X 1 → P with αf = h. The conditions for a module X 1 to be X 2subprojective are given in [15,20]. For any X ∈ R − M od, we denote by Pr −1 (X) the class {L ∈ R − M od : X is L-subprojective}.…”