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

A module is extending (or CS, C1) if every complement submodule is a direct summand or,
equivalently, every submodule is essential in a direct summand. The terms extending and …

In a recent paper by Wang and Ding, it was stated that any ring which is generalized
supplemented as a left module over itself is semiperfect. The purpose of this note is to show …

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 …

In this work generalized supplemented semimodules are defined which generalize
generalized supplemented modules. We investigate some properties of these semimodules …

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 …

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

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M
has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic …

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 …

Let $ R $ be a ring and $ M $ be a left $ R $-module. $ M $ is called generalized $\oplus $-
supplemented if every submodule of $ M $ has a generalized supplement that is a direct …