In this paper, we study a module which is lifting and supplemented relative to a module
class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules …

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 …

Let R be a ring with identity and let M be a left R-module. M is called µ-lifting modulei f for
every sub module A of M, There exists a direct summand D of M such that M= D⊕ D', for …

A module M is said to be lifting, if it satisfies the following lifting property: For any submodule
X of M, there exists a direct summand of M which is a co-essential submodule of X in M. This …

The notions of t-lifting modules and t-small submodules of a module M were introduced in
Amouzegar et al.(Vietnam J. Math. 42, 159–169, 2014). But the authors restricted the …

The notion of⊕-supplemented modules is a proper generalization of lifting modules. As a
proper generalization of⊕-supplemented modules, we introduce (D 12)-modules and obtain …

Lifting modules as a main concept in module theory have been studied and investigated
extensively in recent decades. The first author in [1] tried to consider and investigate this …

We say that a module M is weak lifting if M is supplemented and every supplement
submodule of M is a direct summand. The module M is called lifting, if it is weak lifting and …

A module is extending relative to a module class in two different ways. The definition and
general properties of such extending modules have been given in [3] and also in the case of …

We introduce the concepts of lifting modules and (quasi‐) discrete modules relative to a
given left module. We also introduce the notion of SSRS‐modules. It is shown that (1) if M is …