ABSTRACT A module M is said to be square free if whenever its submodule is isomorphic to
N 2= N⊕ N for some module N, then N= 0. Dually, a module M is said to be d-square free …

In this paper, we first introduce the concept of “factor-square-full” modules defined as
follows: For any proper submodule X of M, there exist a proper submodule Y of M with X⊆ Y …

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 …

We show that if $ M $ is a Utumi module, in particular if $ M $ is quasi-continuous, then $ M=
Q\oplus K $, where $ Q $ is quasi-injective that is both a square-full as well as a dual-square …

A module M is called dual automorphism invariant if whenever X 1 and X 2 are small
submodules of M, then each epimorphism f: M/X 1→ M/X 2 lifts to an endomorphism g of M …

A module M is said to be dual square free if, for any nonzero module N, there are no
epimorphisms from M to N2. Oppositely, a module M is called dual square full if, for any …

Модуль M называется дуально автоморфизм-инвариантным, если любой малый
эпиморфизм f: M/X1→ M/X2 поднимается до эндоморфизма g модуля M для любых …