Abstract
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. Algebra37 (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.
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