Given modules M-R and A(R), M-R is said to be absolutely A(R)-pure if A circle times M -> A circle times B is a monomorphism for every extension B-R of M-R. For a module AR, the absolutely pure domain of A(R) is defined to be the collection of all modules M-R such that M-R is absolutely A(R)-pure. As an opposite to flatness, a module A(R) is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and of f-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.