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

Let R be a ring and M a right R-module. It is shown that:(1) M is Artinian if and only if M is a
GAS-module and satisfies DCC on generalized supplement submodules and on small …

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 …

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 …

In this work we will attempt to define and investigate new classes of modules named⨁-
gsupplemented and⨁-g-radical supplemented as a proper generalization of class of g-lifting …

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 …

In the present paper, we define generalized (amply) cofinitely supplemented modules, and
generalized⊕-cofinitely supplemented modules are defined as a generalization of (amply) …

Abstract Motivated by [1], we study on\tau-lifting modules (rings) and\tau-semiperfect
modules (rings) for a preradical\tau and give some equivalent conditions. We prove that; i) if …