In this paper, we define and study the
-subinjectivity domain of a module
M where
is a complete cotorsion pair, which consists of those modules
N such that, for every extension
K of
N with
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In this paper, we define and study the
-subinjectivity domain of a module
M where
is a complete cotorsion pair, which consists of those modules
N such that, for every extension
K of
N with
in
, any homomorphism
can be extended to a homomorphism
. 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
-indigent modules—that is, the modules whose
-subinjectivity domains are as small as possible—for the cotorsion pair
, where
is the class of FP-injective modules. Additionally, we determine the rings for which all (simple) right modules are either
-indigent or FP-injective. We further investigate
-indigent Abelian groups in the category of torsion Abelian groups for the well-known example of the flat cotorsion pair
, where
is the class of flat modules.
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