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.