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Let R be a ring. A collection of R-modules containing the zero module and closed under isomorphisms will be denoted by X. An R-module M is said to be X -lifting if for every X-submodule N of M there exists A ≤ N such that M = A ⊕ B and N ∩ B is small in B [11]. In the present paper, we consider the question: Can we characterize X-lifting modules via objects of the class X ? [ABSTRACT FROM AUTHOR]

An R-module M is said to be a C1-module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of the τ-C1 module for a hereditary torsion theory τ on Mod-R. τ -C1 modules are a generalization of C1-modules. [ABSTRACT FROM AUTHOR]

This article introduces the concept of a τ-supplemented module as follows: Given a hereditary torsion theory in Mod R with associated torsion functor τ, we say that a module M is τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. We present here some fundamental properties of this class of modules and study the decompositions of τ-supplemented modules under certain conditions on modules. The question of which direct sum of τ-supplemented R-modules are τ-supplemented is treated here. [ABSTRACT FROM AUTHOR]