Subinjectivity Relative to Cotorsion Pairs

In this paper, we define and study the X -subinjectivity domain of a module M where X = ( A , B ) is a complete cotorsion pair, which consists of those modules N such that, for every extension K of N with K / N in A , any homomorphism f : N → M can be extended to a homomorphism g : K → M . This approach allows us to characterize some classical rings in terms of these domains and generalize some known results. In particular, we classify the rings with X -indigent modules—that is, the modules whose X -subinjectivity domains are as small as possible—for the cotorsion pair X = ( FC , FI ) , where FI is the class of FP-injective modules. Additionally, we determine the rings for which all (simple) right modules are either X -indigent or FP-injective. We further investigate X -indigent Abelian groups in the category of torsion Abelian groups for the well-known example of the flat cotorsion pair X = ( FL , EC ) , where FL is the class of flat modules.