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A pure-injective module is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module is called subinjective relative to a module if for every extension of , every homomorphism can be extended to a homomorphism . The subinjectivity domain of the module is defined to be the class of modules such that is -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 simple, uniform, or indecomposable pure-injective module is injective or subinjective relative only to the smallest possible family of modules is investigated.