Search

In this article, we introduce a generalization of quasi-discrete (a GQD-module) by using the notion ofH-supplemented modules and investigate some properties of GQD-modules. First we consider some properties of a relative radical projectivity which is useful in analyzing the structure ofH-supplemented modules. We apply them to the study of direct sums of GQD-modules. Moreover, we prove that anyH-supplemented (lifting) module with finite internal exchange properly (FIEP) has an indecomposable decomposition and show that, for anH-supplemented (lifting) module, the finite exchange property implies the full exchange property. [ABSTRACT FROM PUBLISHER]

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]