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A module M is called N -almost-invariant for a module N if, for any homomorphism α : F → E , either α (N) ⊆ M , or there exist nonzero direct summands F ′ of F and E ′ of E such that α | F ′ : F ′ → E ′ is an isomorphism and (α | F ′ ) − 1 (M ∩ E ′) ⊆ N ∩ F ′ , where E and F are the injective hulls of M and N , respectively. This is a generalization of an almost N -injective module. In this paper, we give a new characterization of a generalized N -injective module by homomorphisms between their injective hulls, and consider conditions for an N -almost-invariant module to be almost N -injective. Moreover, we study a relationship between generalized N -injective, almost N -injective and N -almost-invariant modules. [ABSTRACT FROM AUTHOR]

A module M is said to be lifting if, for any submodule X of M , there exists a decomposition M = A ⊕ B such that A ⊆ X and X / A is a small submodule of M / A. A lifting module is defined as a dual concept of the extending module. A module M is said to have the finite internal exchange property if, for any direct summand X of M and any finite direct sum decomposition M = M 1 ⊕ ⋯ ⊕ M n , there exists a direct summand N i of M i (i = 1 , ... , n) such that M = X ⊕ N 1 ⊕ ⋯ ⊕ N n . This paper is concerned with the following two fundamental unsolved problems of lifting modules: "Classify those rings all of whose lifting modules have the finite internal exchange property" and "When is a direct sum of indecomposable lifting modules lifting?". In this paper, we prove that any d -square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property. [ABSTRACT FROM AUTHOR]

A module M is said to be H-supplemented if, for any submodule X of M, there exists a direct summand M′ of M such that M = X + Y if and only if M = M′ + Y for all Y ⊆ M (cf. [S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, Vol. 147 (Cambridge University Press, 1999)]). We say that a module M is semi-lifting if any direct summand of M is H-supplemented. A H-supplemented module is a dual to a Goldie-extending module which was introduced by Akalan-Birkenmeier-Tercan [Goldie extending modules, Comm. Algebra 37 (2009) 663-683]. In this paper, we give some characterizations of semi-lifting modules and H-supplemented modules. In addition, we consider generalizations of relative projectivities and apply them to the study of direct sums of semi-lifting modules. [ABSTRACT FROM AUTHOR]