Extending modules are generalizations of injective modules and, dually, lifting modules
generalize projective supplemented modules. There is a certain asymmetry in this duality …

We introduce and investigate e-small and s-essential submodules, and prove that Rad<
sub> e(M)=∩{N⊴ M/N is maximal in M}=∑{N⊆ M/N≪< sub> e M} and Soc< sub> …

In this paper, δ-lifting and δ-supplemented modules are defined as generalizations of lifting
and supplemented modules. Several properties of these modules are proved. New …

Let I be an ideal of a ring R. Consider the following conditions on R: 1. If X is a finitely
generated submodule of a finitely generated projective module P, then X= A⨁ B where is a …

In this paper, principally supplemented (𝛿-supplemented), and principally lifting (𝛿-lifting)
semimodules are defined as generalizations of principally supplemented (𝛿-supplemented) …

In present work, we describe and investigate torsion theoretic versions of δ‐supplemented
modules via a hereditary torsion theory τ. With this aim, first, we define δτ‐small …

The concept of Hopfian modules has been extensively studied in the literature. In this paper,
we introduce and study the notion of δ-weakly Hopfian modules. The class of δ-weakly …

Let R be a ring and M a right R‐module. It is shown that (1) δ (M) is Noetherian if and only if
M satisfies ACC on δ‐small submodules;(2) δ (M) is Artinian if and only if M satisfies DCC on …

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a
class of modules which is an analogous to δsupplemented modules and principally⊕ …

Zhou defined\delta-semiperfect rings as a proper generalization of semiperfect rings. The
purpose of this paper is to discuss relative notions of supplemented modules and to show …