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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]

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]