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We consider two types of weak-injectivity domains of a module. Small weak-injectivity domains gauge the extent of weak injectivity of modules; weak-injectivity domains are an alternative way to gauge their injectivity. We focus on notions which are opposite, respectively, to weakly injective or injective modules according to these two schemes. While we do consider both notions to some extent, our emphasis is on modules with smallest weak-injectivity domain; we name those modules extremely poor. The intersection of all (small) domains of weak injectivity is the natural focus of attention. The intersection of all (small) domains of weak injectivity contains the class of all (finitely generated) semisimple modules, but it is not always equal to it. The domain of weak injectivity of an extremely poor module thusly depends on the ring in question. We consider necessary and sufficient conditions for said intersections to …