  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 \to M can be extended to a homomorphism
K \to 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.
