Projectivity relative to closed (neat) submodules

An [Formula: see text]-module [Formula: see text] is called closed (neat) projective if, for every closed (neat) submodule [Formula: see text] of every [Formula: see text]-module [Formula: see text], every homomorphism from [Formula: see text] to [Formula: see text] lifts to [Formula: see text]. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right [Formula: see text]-module is closed (neat) projective is studied. Furthermore, the relationship among the proper classes which are induced by closed submodules, neat submodules, pure submodules and [Formula: see text]-pure submodules are investigated.

α-modules and generalized submodules

A QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable.

Strict Mittag–Leffler modules over Gorenstein injective modules

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

Some Results on Pure Submodules Relative to Submodule

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

On h-purifiable submodule of QTAG-module

Different concepts and decomposition theorems have been done for QTAG-modules by a number of authors. The concept of quasi $h$-pure submodules were introduced and different characterizations were obtained in \cite{5}. %[5]. The purpose of this paper is to obtain the relation between purifiability of a submodule and quasi $h$-pure submodules. Further we obtained results which shows that purifiability of a submodule is very much dependent on the purifiability of a $h$-pure and $h$-dense submodule of the given submodule.

On Strongly F – Regular Modules and Strongly Pure Intersection Property

A submoduleA of amodule M is said to be strongly pure , if for each finite subset {ai} in A , (equivalently, for each a ?A) there exists ahomomorphism f : M ?A such that f(ai) = ai, ?i(f(a)=a).A module M is said to be strongly F–regular if each submodule of M is strongly pure .The main purpose of this paper is to develop the properties of strongly F–regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .