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We prove that a module M is cofinitely weak supplemented or briefly cws (i.e., every submodule N of M with M=N finitely generated, has a weak supplement) if and only if every maximal submodule has a weak supplement. If M is a cws-module then every M-generated module is a cws-module. Every module is cws if and only if the ring is semilocal. We study also modules, whose finitely generated submodules have weak supplements.

Abstract. In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to ⊕ p∈P Zp, where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of U (N) , where U ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group M , it is shown that M can not be torsion, and each p-primary component of M is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa.

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In a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N -subinjective if every homomorphism N → M extends to some E(N ) → M , where E(N ) is the injective hull of N . Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J 2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized.