A study of summable QTAG‑modules

This manuscript deals with the quasi-isomorphic invariants for QTAG modules; specially the cases when the module is summable, 𝜎 -summable, ( 𝜔 + n ) -projective or HT -module. We show that if for a QTAG module M with a submodule N such that M / N is bounded, then M is weakly 𝜔 1 -separable if and only if N is weakly 𝜔 1 -separable .


Introduction and preliminaries
The concepts for groups such as projectivity, purity, height etc carried to modules preserving the sense. For getting those results of groups, which does not hold for modules; some constraints applied on the structure of module or underlying ring. After taking into effects these constraints for the QTAG-module structure, numerous natural facts of groups can be established that are not true generally. The findings in [10] serve as the inspiration for the outcomes in this work.
The study on the structure of QTAG-modules was started by Singh [11]. After that many researchers such as Khan, Mehdi, Abbasi etc. generalized different concepts of groups to QTAG-modules [4,9] etc. They introduced various notions and structures for QTAG-modules motivated from group structure and obtained some exciting results. Yet many concepts remain to generalize for modules.
Some of the fundamental definitions used in this manuscript have already appeared in one of the co-authors' previous works; these are offered as quotations and are duly cited here. "A module M over an associative ring R with unity is a QTAG -module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules [12]. All the rings R considered here are associative with unity and modules M are unital QTAG -modules. An element x ∈ M is uniform, if xR is a non-zero uniform (hence uniserial) module and for any R-module M with a unique composition series, d(M) denotes its composition length. For a uniform element x ∈ M, e(x) = d(xR) and (iii) Given any N ∈ N and any countable subset X of M, there exists K ∈ N containing N ∪ X , such that K/N is countably generated [4].
Every submodule in a nice system is nice submodule. A h-reduced QTAG -module M is called totally projective if it has a nice system and direct sums and direct summands of totally projective modules are also totally projective.

Main results
Mehdi et al. defined quasi isomorphic modules [5], studied their properties and quasi-isomorphic invariants. First we recall the definitions of ( + n)-projective, HT-modules, 1 -separable modules. "A QTAG -module M is ( + n)-projective if and only if there exists a submodule N ⊂ H n (M) such that M/N is a direct sum of uniserial modules. M is a HT-module [6] if every homomorphism from M to N is small, where N is a direct sum of uniserial modules. Equivalently M is a HTmodule if and only if N ⊃ Soc(H k (M)) for some k < whenever M/N is a direct sum of uniserial modules. A separable QTAG -module M is weakly 1 -separable [1] if and only if for every countably generated submodule N ⊂ M , its closure N is also countably generated i, e.
. Two QTAG modules Here we investigate the modules M and M ′ when H k (M) is isomorphic to a submodule of M ′ . Equivalently we consider the submodule N ⊂ M and H k (M) ⊆ N. Now we are able to prove the following:  H (N) . Since M and H n (M) are summable they are Σ-modules as they are fully invariant submodules [8]. Therefore N is also a Σ -module. Let K be a high submodule of N.
T h e r e f o r e For the converse suppose N is summable. Now N is a Σ -module and by [5,7]; M is a Σ -module. Since N is summable H (N) = H (M) is summable, therefore M is also summable. ◻

Theorem 2 Let M be a QTAG -module with a submodule N such that H n (M) ⊆ N for some n < . Then N is -summable if and only if M is -summable.
Proof If M is -summable, then by [1,8]; Proof Suppose N is ( + n)-projective. Now there exists a submodule K ⊆ H n (N) such that N/K is a direct sum of uniserial modules. Therefore then being a submodule of M, N is also ( + n)-projective [3].
, a direct sum of uniserial modules, we have is bounded. Therefore, is a direct sum of uniserial modules as K ⊆ H k (M) and the result follows. ◻ For the QTAG modules M, M ′ a homomorphism f ∶ M ⟶ M � is said to be small if ker f contains a large submodule. Naji [6] defined HT-modules with respect to a family F of QTAG -modules. A QTAG -module is a HT-module with respect to a family F of QTAG -modules if Hom(M, K ) = Hom s (M, K ) for every K ∈ F , whenever K is a direct sum of uniserial modules. This can be stated as follows: Definition 1 [6] A QTAG -module M is said to be a HT-module if every homomorphism f ∶ M ⟶ M � is small whenever M ′ is a direct sum of uniserial modules.
Since these definitions are equivalent we are able to prove the following: . Therefore M is separable if and only if N is separable. Since the property of being 1 -separable is inherited by the submodules, N is also weakly