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

Let t be a radical for the category of left R-modules for a ring R. If M is a t-coatomic module,
that is, if M has no nonzero t-torsion factor module, then t (M) is small in M. If V is a t …

Proper classes of monomorphisms and short exact sequences were introduced by
Buchsbaum to study relative homological algebra. It was observed in abelian group theory …

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism
Y→ M is neat in Y, ie, the induced map Hom (S, Y)→ Hom (S, M) is surjective for any simple …

Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-
supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad …

In this paper, we introduce some new results about the relationship between two concepts
namely singular and essential and c-ring. We prove that if any module 𝑊 is a direct sum of …

In this paper; we will study the relationship between simple submodules of M and the C-ring.
A ring R is called C-ring if every torsion module M over R has simple submodules. We …

In this thesis we investigate proper classes of short exact sequences, especially those
proper classes induced by supplement-like and complement-like submodules in the …

Over an arbitrary ring, a module M is said to be Z∗ Z^ ∗-semilocal if every submodule U of M
has a Z∗ Z^ ∗-supplement V in M, ie, M= U+ V and U∩ V⊆ Z∗ V, U ∩\kern0. 5em V ⊆ Z^ ∗ …

In this survey article, we shall give some properties of classes of short exact sequences of
modules related with complement and supplement submodules; some of them form proper …