F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172218.1 Research Article Articles Endo-Greatly Bounded Modules

[version 1; peer review: 1 not approved]

 kamel Hussein sattar Conceptualization Formal Analysis Funding Acquisition Investigation Methodology Resources Software Validation Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0009-0003-5491-9776 a 1 Najad Shihab Buthyna Supervision 1 Department of Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, Iraq sattar.kamel2203@ihcoedu.uobaghdad.edu.iq

No competing interests were disclosed.

 11 12 2025 2025 14 1392 3 12 2025 Copyright: © 2025 kamel Hussein s and Najad Shihab B 2025 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

In the present paper, assuming every module H is unitary and every ring S is commutative with identity, we address the concepts of bounded modules and Endo-R.B.modules. We provide some examples, corollaries, remarks, and properties our new concept. An Endo-R.B S-module is considered to be more powerful than bounded module and Endo-R.B module that will present both of them in some details in the introduction. Motivated by these notions, we introduce and discuss a new class of modules called the Endo-Greatly Bounded module (abbreviated as Endo-G.B. module), a concept which has not been previously reported in the literature.

 Methods

We formally define the Endo-G.B. module and provide examples, corollaries, and properties to illustrate this new concept. The study employs scalar modules and fully invariant submodules as crucial tools to establish connections. It must be emphasized that scalar modules and prime played a major role in achieving new results in this paper. We specifically investigate the important relationships between an Endo-G.B module and prime module as well as their generalizations.

 Results

We get and prove that every Endo-G.B. module is Bounded, where as the converse is not generally true. Furthermore, we establish new results connecting our new class of S-modules to cyclic bounded modules, projective modules, multiplication modules, and finitely annihilated modules. We also derive necessary and sufficient conditions for an Endo-G.B. module to exhibit specific flexible properties.

 Conclusions

In this paper, we presented a new class of S- modules called Endo-greatly bounded modules, these findings highlight the depth and flexibility of Endo-G.B. modules with in modern algebraic frameworks. This study opens new directions for further research in module theory, particularly in exploring the interaction between bounded ness and endomorphism structures, and may motivate future algebraic investigations in commutative algebra.

 Endo-Greatly Bounded module scalar module finitely annihilated module The author(s) declared that no grants were involved in supporting this work. 1. Introduction

Let S be a commutative ring with identity and Η aunitary (left) S-module. The concept of bounded module in term of it is annihilator, where an S-module Η is called bounded if ∃ x ∈ Η such that 1 ann S ( x ) = ann S ( H ) .

In addition, the concept was studied in detail by Al-Ani, 2 and some necessary conditions were investigated to connect the bounded module with other types of modules and submodules. In addition, gave the definition of bounded submodule, where a submodule N is called bounded if ∃ y ∈ N such that ann s ( y ) = ann s ( N ) as well as the comprehensible of fully Bounded module is studied.

An S-module Η is called fully bounded if each proper submodule N of Η is bounded. 3

The goal of this work is to introduce a new type of module that has not been recognized before, and is much more powerful than abounded module called an Endo-G.B. module. Throughout this paper, we point out certain analogies between Endo-G.B. and other types of S-modules. In the first section, we provide a definition of the Endo-G.B. submodule, and the Endo-G.B. module will be presented with some properties and examples. In section two, we explain The relationship between bounded and finitely annihilated with an Endo-G.B. module at the end, prime and scalar modules are involved in this paper to discover their connection with Endo-G.B. modules, S-module Η is called scalar if for all φ ∈ End ( Η ) , ∃ r ∈ S such φ ( x ) = rx , ∀ x ∈ H . 4 Buthyna Shihab introduced the concept of almost bounded, where submodule A is called almost bounded if 5 ∃ x ∈ Η , x ∉ A , such that ann s ( x ) = ann s ( A ) .

Endo-G.B. is a stronger concept than the Endo-R.B. module. 6

An S-module Η is called prime if for all submodule A of Η we have 7 ann s ( A ) = ann s ( H ) .

Also, an S-module Η is called to be finitaly annihilated S-module if ∃ a finitaly generated submodule N of Η like that ann s ( N ) = ann s ( H ) . 8

That is clear all prim module is bounded and finitely annihilated module. An S-module Η is called to be multiplication if for all submodule N of Η there exist an ideal I of S such that IΗ = N . 9

2. Endo-greatly bounded modules

In this section, we first provide a definition of Endo-G.B. modules with some examples to introduce the concept of the End-G.B. module. In addition, in this section, many properties and corollaries are proven.

2.1 Definition

An S -module Η is called Endo- Greatly Bounded module if every proper submodule of Η is an Endo- Greatly Bounded module.

Z p as a Z p –module is an Endo-GB module,where p is the prime number.

Since < 0 ¯ > is the only one proper submodule of z p , if we take x ¯ = 2 ¯ ∈ Z 3

Define φ , ψ : Z 3 → Z 3 by φ ( x ¯ ) = 3 x ¯ ∀ x ¯ ∈ Z 3 ψ ( x ¯ ) = x ¯ ∀ x ¯ ∈ Z 3 φ ( x ¯ ) = φ ( 2 ¯ ) = 3 . 2 ¯ = 0 ¯ ∈ < 0 ¯ > ann z ( ( φ oψ ) ( x ¯ ) ) = ann z ( φ ( ψ ( x ¯ ) ) . = ann z ( φ ( x ¯ ) ) = ann z ( 3 x ¯ ) = ann z ( 3 .2 ¯ ) = ann z ( 0 ¯ ) = Z = ann z ( < 0 ¯ > )

Therefor Z p is an Endo-G.B. module.

Let Η = Z 4 ⊕ Z 2 as a Z-module, Define φ , ψ : Η → Η as φ ( x ¯ , y ¯ ) = ( 2 ¯ , 0 ¯ ) ∀ ( x ¯ , y ¯ ) ∈ Η ψ ( x ¯ , b ¯ ) = ( x ¯ , y ¯ ) ∀ ( x ¯ , y ¯ ) ∈ Η

If we take N = < 2 ¯ > ⊕ Z 2 and ( x ¯ , y ¯ ) = ( 3 ¯ , 0 ¯ ) ∈ Η , then φ ( 3 ¯ , 0 ¯ ) = ( 2 ¯ , 0 ¯ ) ∈ N and ψ ( 3 ¯ , 0 ¯ ) ∉ N ( φoψ ) ( 3 ¯ , 0 ¯ ) = φ ( ψ ( 3 ¯ , 0 ¯ ) ) = φ ( 3 ¯ , 0 ¯ ) = ( 2 ¯ , 0 ¯ ) , therefor 2 Z = ann z ( N ) ≠ ann z ( φ ( 3 ¯ , 0 ¯ ) ) = ann z ( 2 ¯ , 0 ¯ ) = 2 Z

Hence, N is the Endo-GB submodule, which implies that Η is not an Endo-GBmodule.

: Let Z 36 as a Z-module be not End-G.B. S-module. If we take N = < 12 ¯ > and x ¯ = 9 ¯ ∈ Z 36 .

Define φ ( x ¯ ) = 4 x ¯ , ∀ x ¯ ∈ Z 36 , ψ ( x ¯ )= 2 x ¯ ∀ x ¯ ∈ Z 36 φ ( x ¯ ) = ψ ( 9 ¯ ) = 4 ( 9 ¯ ) = 36 ¯ = 0 ¯ ∈ N , ψ ( x ¯ ) = ψ ( 9 ¯ ) = 2 ( 9 ¯ ) = 18 ¯ ∉ N ( φoψ ) ( x ¯ ) = ( φoψ ) ( 9 ¯ ) = φ ( ψ ( 9 ¯ ) ) = φ ( 18 ¯ ) = 4 ( 18 ¯ ) = 72 ¯ = 0 ¯ ann ( φoψ ) ( x ¯ ) = ann ( 0 ¯ ) = Z , but ann ( N ) = ann < 12 ¯ > = 3 Z ≠ ann ( ( φoψ ) ( x ¯ ) ) = Z .

3. Results

Every Endo-G.B. module is bounded, but the opposite is not generally true; for example:

Let Η = Z 2 ⊕ Z 2 as a Z-module and N = Z 2 ⊕ < 0 ¯ > , x ¯ = ( 0 ¯ , 1 ¯ ) ∈ Η

Let φ , ψ : Η → Η as φ ( x ¯ ) = φ ( a ¯ , b ¯ ) = ( 0 ¯ , 0 ¯ ) ∀ ( a ¯ , b ¯ ) ∈ Η ψ ( x ¯ ) = ψ ( a ¯ , b ¯ ) = ( a ¯ , b ¯ ) ∀ ( a ¯ , b ¯ ) ∈ Η

Then ann z ( N ) = 2 Z and ann z ( ( φoψ ) ( 0 ¯ , 1 ¯ ))= ann z ( 0 ¯ , 0 ¯ ) = Z

Therefor Η is not Endo-G.B module, but ann ( x ¯ ) = ann ( 0 ¯ , 1 ¯ ) = 2 Z = ann ( Η ) = 2 Z hence Η is bounded module.

All proper submodule of Endo-G.B. module is also Endo-G.B.

As a direct result from no. (2) we get the following:

The intersection of two Endo-G.B. submodules of Η is an Endo-G.B.

Since if N 1 , N 2 ≤ Η and N 1 , N 2 are two Endo-G.B. submodules, then N 1 ∩ N 2 ⊆ N 1 and N 1 ∩ N 2 ⊆ N 2 which implies that by no (2) N 1 ∩ N 2 is Endo-G.B. submodule.

Let Η be an Endo-G.B.S-module and A 1 , A 2 ≤ Η such that A 1 ⊆ A 2 or A 2 ⊆ A 1 . Then A 1 ∪ A 2 is Endo-G.B. submodule.

Let Η be an Endo-G.B.S-module and B ≤ Η . Then Η ∕ B is an Endo-G.B.S-module.

Let A , B ≤ Η and A containing B such that A / B ≤ Η / B .

Since Η is an Endo-G.B. module. Then A is an End-G.B. submodule that is, there exists φ , ψ ∈ End ( Η ) such that either φ ( x ) ∈ N or ψ ( x ) ∈ N , and ann S ( φoψ ( x ) ) = ann S ( A ) .

Let φ , ψ : Η B → Η B by φ ( x + B ) = φ ( x ) + B , ∀ x ∈ Η and ψ ( x + B ) = ψ ( x ) + B , ∀ x ∈ Η , then it , s clear that φ , ψ are well –defined and an endomorphism of Η / B .

Obviously, ψ ( x + B ) ∈ A / B so ann R ( A B ) ⊆ ann S ( φo ψ ( x + B ) ) .

Now, let r ∈ ann S ( φoψ ( x + B ) ) that means r ∈ ann S φ ( ψ ( x + B ) ) = rφ ( ψ ( x ) + B ) = r ( φoψ ( x ) + B ) = B , ∀ x ∈ Η . r ( φoψ ( x + B ) ) = B = r ( φ [ ψ ( x + B ) ] ) = B r ( φ [ ψ ( x ) + B ] ) = B = r ( φ ( ψ ( x ) + B ) ) = B r ( φψ ( x ) + B ) = B r φψ ( x ) + B = B .

Since rφψ ( x ) ∈ rA , ∀ x ∈ Η , therefor rN + B = B ∴ r ( A + B ) = B .

We conclude that r ∈ ann S ( A B ) .

Let Η 1 , Η 2 be two S -modules such that ann S ( Η 1 ) + ann S ( Η 2 ) = S .

Then Η 1 ⨁ Η 2 is an Endo-G.B.S-module if and only if Η 1 and Η 2 are Endo-G.B. S-modules.

Assume that Η 1 ⨁ Η 2 is an Endo-G.B, the A 1 ⨁ A 2 ≤ Η 1 ⨁ Η 2 where

A 1 ≤ Η 1 and A 2 ≤ Η 2 .

Let φ , ψ : Η 1 ⨁ Η 2 → Η 1 ⨁ Η 2 be an endomorphism of Η 1 ⨁ Η 2 define as

either φ ( a , b ) = r ( a , b ) , like that φ ( a , b ) ∈ A 1 ⨁ A 2

or ψ ( a , b ) = r ( a , b ) , like that ψ ( a , b ) ∈ A 1 ⨁ A 2

So that , ( ra , rb ) ∈ A 1 ⨁ A 2 where ra ∈ A 1 , rb ∈ A 2 and ann S ( φoψ ( a , b ) ) = ann S ( A 1 ⨁ A 2 ) .

Consider the composition Η 1 → i Η 1 ⨁ Η 2 → φ Η 1 ⨁ Η 2 → ψ Η 1 ⨁ Η 2 → j Η 1 .

Let a ∈ Η 1 , then ( joψoφoi ) ( a ) = joψoφ ( i ( a ) ) = joψoφ ( a , 0 ) = joψ ( φ ( a , o ) ) = joψ ( ra , o ) = j ( ψ ( ra , 0 ) ) = j ( ra , 0 ) = ra ∈ A 1 .

Now , to show that ann S ( joψoφoi ( a ) ) = ann S ( A 1 ) .

Let t ∈ ann S ( joψoφoi ( a ) ) , then t ( joψoφoi ( a ) ) = 0 , that is t ( ra ) = 0 ,

Where t ∈ S . This implies that t ∈ ann S ( A 1 ) .

Hence ann S ( joψoφoi ( a ) ) = ann S ( A 1 ) .

So Η 1 is an End. G.B. Similarly,we can prove that Η 2 is the Endo-GB S-module.

Conversely, suppose that Η 1 and Η 2 are Endo-GB S-modules. Let A 1 and A 2 are two submodule of Η 1 , Η 2 respectively. Then ∃ φ 1 , ψ 1 , φ 2 , ψ 2 ∈ Endo ( Η 1 ) ,

Endo ( Η 2 ) respectively defined as:

φ 1 , ψ 1 : Η 1 → Η 1 such that φ 1 ( a ) = ra , ψ 1 ( a ) = a ∀ a ∈ Η 1

φ 2 , ψ 2 : Η 2 → Η 2 such that φ 2 ( b ) = rb , ψ 2 ( b ) = b ∀ b ∈ Η 2

Also, φ 1 ( a ) = ra ∈ A 1 , ψ 1 ( a ) = a ∈ A 1 and φ 2 ( b ) = rb ∈ A 2 , ψ 2 ( b ) = b ∈ A 2

where ann S ( ( φ 1 o ψ 1 ) ( a ) ) = ann S ( A 1 ) , ann S ( ( φ 2 o ψ 2 ) ( b ) ) = ann S ( A 2 )

Now, let f ( φ 1 , φ 2 ) and g ( ψ 1 , ψ 2 ) , where f , g : Η 1 ⨁ Η 2 → Η 1 ⨁ Η 2 , defined as fog = ( φ 1 o ψ 1 , φ 2 o ψ 2 ) ( a , b ) = ( φ 1 o ψ 1 ( a ) , φ 2 o ψ 2 ( b ) ) = φ 1 ( ψ 1 ( a ) ) , φ 2 ( ψ 2 ( b ) )

= ( φ 1 ( a ) , φ 2 ( b ) ) = ( ra , rb ) ∈ A 1 ⨁ A 2 where A 1 ⨁ A 2 ≤ Η 1 ⨁ Η 2 .

We claim that ann S ( fog ( φ 1 o ψ 1 , φ 2 o ψ 2 ) ) = ann S ( A 1 ⨁ A 2 ) ,

Let t ∈ ann S ( fog ( φ 1 o ψ 1 , φ 2 o ψ 2 ) ) implies that t ( fog ( φ 1 o ψ 1 , φ 2 o ψ 2 ) ( a , b ) ) = ( 0 , 0 )

= t ( φ 1 o ψ 1 ( a ) , φ 2 o ψ 2 ( b ) ) = t ( ra , rb ) .

Thus, t ∈ ann S ( A 1 ⨁ A 2 ) .

If Η 1 and Η 2 are two Endo-G.B.S-modules, then it is not necessary that Η 1 ⨁ Η 2 be an Endo-G.B. module so the condition in Proposition (3.6) can not be dropped and the next example shows that:

Let Η 1 = Z 6 and N 1 = ( 3 ¯ ) . Define φ 1 , ψ 1 : Z 6 → Z 6 as φ 1 ( a ¯ ) = 3 a , ψ 1 ( a ¯ ) = a ¯ . Then ann S ( φ 1 o ψ 1 ) ( a ¯ ) = ann S ( N 1 ) , φ 1 ( a ¯ ) ∈ N 1 and

let M 2 = Z 3 , N 1 = ( 0 ¯ ) .

Define φ 2 , ψ 2 : Z 3 → Z 3 as φ 2 ( a ¯ ) = 0 ¯ , ψ 2 ( a ¯ ) = a ¯

Then ann S ( φ 2 O ψ 2 ( a ¯ ) ) = ann S ( N 2 ) , φ 1 ( a ¯ ) ∈ N 2 .

Let Η = Z 6 ⨁ Z 3 and N = ( 3 ¯ ) ⨁ ( 0 ¯ ) . Define φ , ψ : Z 6 ⨁ Z 3 → Z 6 ⨁ Z 3 as φ ( a ¯ , b ¯ ) = ( a ¯ , 0 ) . Then Η is not Endo-G.B. modul since φ ( a ¯ , b ¯ ) ∉ N .

Let n be a positive integer and Η i ( i ∈ I ) be an S-module, such that ann S Η i + ann S Η j = S , ∀ 1 ≤ i ≤ j ≤ n . Then Η = Η i ⨁ Η j ⨁ … … … … . ⨁ Η n is an Endo-GB module if and only if Η i is Endo-G.B. module ∀ 1 ≤ i ≤ n .

The proof by induction and by using Proposition (3.6).

The direct summand of the Endo-G.B. module Η is also Endo-G.B.

Assume that A is a direct summand of Η , then Η = A ⨁ B for some sub module B of Η .

Since A ≤ A ⨁ B , by using the hereditary property of an Endo-G.B. module we have That A is an Endo-G.B. submodule.

Let Η = A ⨁ B be an S-module, where A and B are two sub modules of Η such that ann S ( A ) + ann S ( B ) = S . Then A and B are End-G.B. submodules if and only if Η is End-G.B.S-module.

Suppose that Η is Endo-G.B.S-module. Since A and B are direct summands of Η , then by Remark (3.10) A and B are Endo-G.B. submodules.

Conversely, assume that A and B are Endo-GB S-submodules. Let L ⨁ N be a submodule of A ⨁ B where L ≤ A and N ≤ B and ann S ( A ) + ann S ( B ) = S .

Thus, from Proposition (3.6), we find that A ⨁ B is an Endo-G.B.S-module.

Recall that a sub-module A of S-module Η is said to be fulley invariant if φ ( A ) ⊆ A for every φ , ψ ∈ End ( H ) . 10

Let X , Y be two fully invariant sub modules of an S-module Η like that Η = X ⨁ Y . Then X and Y are two Endo-G.B. submodules if and only if Η is End-G.B.S-module.

Assume that Η is an Endo-GB S-module. The direct summands of the Endo-GB module are also Endo-G.B.

Therefore, we conclude that X and Y are Endo-GB S-submodules.

Conversely, suppose that X and Y are Endo-G.B.S-submodules of an S-module Η .

Then in order to show that Η is an Endo-G.B.S-module, Let φ , ψ : Η → Η be an endomorphism and N ≤ Η . Firstly, we have to prove that either φ ( x ) ∈ N or ψ ( x ) ∈ N , where x ∈ Η . Since X , Y are fully invariant S-modules, then N = ( N ∩ X ) ⨁ ( N ∩ Y ) either ψ ( x ) ∈ ( N ∩ X ) ⨁ ( N ∩ Y ) , then ψ ( x ) = φ 1 ( c ) + φ 2 ( d ) or φ ( x ) ∈ ( N ∩ X ) ⨁ ( N ∩ Y ) , then φ ( x ) = ψ 1 ( c ) + ψ 2 ( d ) .

Where φ 1 , ψ 1 , φ 2 , ψ 2 ∈ End ( Η ) either φ 1 ( c ) or ψ 1 ( c ) ∈ N ∩ X , either φ 2 ( d ) or ψ 2 ( d ) ∈ N ∩ Y implies that φ 1 ( c ) , ψ 1 ( c ) ∈ N , φ 1 ( c ) , ψ 1 ( c ) ∈ X and φ 2 ( d ) , ψ 2 ( d ) ∈ N , φ 2 ( d ) , ψ 2 ( d ) ∈ Y .

So, φ 1 ( c ) , ψ 1 ( c ) , φ 2 ( d ) , ψ 2 ( d ) ∈ N which implies ( φ 1 o ψ 1 ) ( c ) + ( φ 2 o ψ 2 ) ( d ) ∈ N , it is clear that ann S ( N ) ⊆ ann S ( φo ψ ( x ) ) .

Now, let r ∈ ann S ( φo ψ ) ( x ) , then r ( φo ψ ) ( x ) = 0 r ( ( φ 1 o ψ 1 ) ( c ) + ( φ 2 o ψ 2 ) ( d ) ) = 0 r ( φ 1 o ψ 1 ) ( c ) = 0 and r ( φ 2 o ø 2 ) ( d ) = 0 r ∈ ann S ( ( φ 1 o ψ 1 ) ( c ) ) = ann S ( X ) and r ∈ ann S ( ( φ 2 o ψ 2 ) ( d ) ) = ann S ( Y ) r ∈ ann S ( N ∩ X ) and r ∈ ann S ( N ∩ Y ) r ( N ∩ X ) + r ( N ∩ Y ) = 0 r [ ( N ∩ X ) + ( N ∩ Y ) ] = 0 r ∈ ann S ( ( N ∩ X ) + ( N ∩ Y ) ) implies that r ∈ ann S ( N ) .

Thus ann S ( ( φo ψ ) ( x ) ) = ann S ( N ) .

The homomorphic image of the Endo-GB module is also an Endo-GB module.

Let Η , Η / be two S-modules, and f : Η → Η / be an S-homomorphism.

Let Η is an Endo-G.B. module, we claim that f ( Η ) is an Endo-G.B. module.

Since Η is an Endo-G.B. there exist φ , ψ ∈ End ( Η ) and define as φ ( x ) = rx , and ψ ( x ) = rx , ∀ x ∈ Η , for all proper submodule N of Η such that:

ann S ( ( φOψ ) ( x ) ) = ann S ( N ) , we know that f ( Η ) ≤ Η / and f ( N ) ≤ f ( Η ) ≤ Η / it , s enough to prove that Η / is an Endo-G.B. module.

Suppose that h , g : Η / → Η / and define as h ( b ) = rf ( a ) , g ( b ) = rf ( a ) , ∀ b ∈ Η /

Then h ( b ) = rf ( a ) = f ( ra ) ∈ f ( N ) where f ( N ) < f ( H ) and g ( b ) = rf ( a ) = f ( ra ) ∈ f ( N ) where f ( N ) < f ( H ) , and ( hog ) ( b ) = ( foφoψ ) ( a ) , ∀ b ∈ Η / and a ∈ Η

Then f ( b ) = φ ( ψ ( a ) ) = φ ( ra ) = ra , ( hog ) ( b ) = f ( φ ( ψ ( ra ) ) ) = f ( φ ( ra ) ) = f ( ra ) ∈ f ( N )

Now, we show that ann S ( ( hog ) ( b ) ) = ann S ( f ( N ) ) .

Let t ∈ ann S ( hog ) ( b ) , then t ∈ ann S ( foφoψ ) ( ra ) ⇒ t ( foφoψ ) ( ra ) = 0 tf ( φ ( ψ ( ra ) ) = 0 ⟹ tf ( φ ( ra ) ) = 0 ⟹ tf ( ra ) = 0

Thus, t ∈ ann S ( Η / ) ⊆ ann S ( f ( N ) ) so ann S ( ( hog ) ( b ) ) ⊆ ann S ( f ( N ) ) , and it , s clear that ann S f ( N ) ⊆ ann S ( hog ) ( b ) . Therefore, Η / is an End-G.B. module implies that f ( Η ) is End-G.B.

4. Endo-G.B. related to bounded and finitely annihilated modules

In section, we study the relationship between the bounded module and Endo-G.B. module, and we will see that an End-G.B. module gives us abounded module directly, but the convers do not work and will provide that with an example, which means an Endo-is stronger than the notion of a bounded module.

Every Endo-G.B.S-module is bounded module.

Assume that φ , ψ : Η → Η as φ ( a ) = a , ψ ( a ) = a , ∀ a ∈ Η , either φ ( a ) ∈ N or ψ ( a ) ∈ N , where N is a proper submodule of Η .

By use the definition of an Endo-G.B. we have ann S ( ( φOψ ) ( a ) ) = ann S ( N ) and implies that ann S ( a ) = ann S ( N ) for all proper sub module N of Η . Hence N is bounded submodule and conclude that every proper submodule is bounded then Η is fully bounded S-module and Η is bounded S-module.

The opposite of ( Remark 4.1) is generally not true in the following example:

Let Η = Z 2 ⨁ Z 2 as Z-module and let N = Z 2 ⨁ ( 0 ¯ ) .

Let φ , ψ : Η → Η by φ ( c ¯ , d ) = ( 0 ¯ , 0 ¯ ) , ψ ( c ¯ , d ¯ ) = ( 0 ¯ , 0 ¯ ) ,

Since ψ ( c ¯ , d ¯ ) = ( 0 ¯ , 0 ¯ ) ∈ N but 2 Z = ann Z ( N ) ≠ ann Z ( 0 ¯ , 0 ¯ ) = Z .

Therefore, Η is not Endo-G.B. while Η is bounded S-module since there exist an element x = ( 0 ¯ , 1 ¯ ) ∈ Z 2 ⨁ Z 2 such that 2 Z = ann Z ( Η ) = ann z ( 0 ¯ , 1 ¯ ) = 2 Z .

Recall an S-module is called cyclic over End ( Η ) if there exist an element x ∈ Η like that for every element h ∈ Η can be writen h = f ( x ) for some f ∈ End ( H ) . 11

Let Η be a bounded S-module and SX is an essential sub module of Η . Then Η is End-G.B.

Let φ , ψ ∈ End ( Η ) and define φ , ψ : Η → Η as φ ( a ) = ra , ψ ( a ) = a , ∀ a ∈ Η . Let N be any proper sub-module of H , then φ ( a ) ∈ N .

Now, let t ∈ ann S ( ( φ 0 ψ ) ( a ) ) , then t ( φ 0 ψ ) ( a ) = 0 , implse that tφ ( a ) = 0 so t ( ra ) = 0 , ∀ r ∈ S . Thus t ∈ ann S ( Sa ) . Since S is commutative ring with identity, then by Lemma (1.2) Alni, we get t ∈ ann S ( a ) = ann S ( Η ) ⊆ ann S ( N ) .

Every Endo-GB S-module is an Endo-GB E-module, where E = End ( Η ) .

Suppose that Η is Endo-G.B.S-module, and let N ≤ Η .

Let φ , ψ ∈ End ( H ) and as φ ( a ) = b , ψ ( a ) = b , ∀ a ∈ Η , so we have φ ( a ) ∈ N and ann S ( ( φOψ ) ( a ) ) = ann S ( N ) . Since N as E-submodule is S-submodule, then φ ( a ) ∈ N as E-submodule.

Now, let δ ∈ ann E ( ( φoψ ) ( a ) ) where δ ∈ End ( Η ) , then δ ( ( φOψ ) ( a ) ) = 0 .

But φ ( a ) = b . Thus δ ( b ) = 0 , ∀ b ∈ N implies that δ ( N ) = 0 , therefore, δ ∈ ann E ( N ) .

Every cyclic submodule of the bounded module is an endo-GB submodule.

Let A be a cyclic sub module of a bounded S-module Η , then there exist x ∈ Η such that A = Sx . Let φ , ψ ∈ Endo ( H ) and as φ ( x ) = rx , ψ ( x ) = x , for every x ∈ Η , φ ∈ End ( Η ) . Then φ ( x ) ∈ A and ann S ( A ) ⊆ ann S ( ( φOψ ) ( x ) ) .

Now, let t ∈ ann S ( ( φoψ ) ( x ) ) , then t ( φOψ ) ( x ) = 0

t φ ( rx ) = 0 , t ( rx ) = 0 , ∀ r ∈ S , t ( Sx ) = 0 .

Thus t ∈ ann S ( Sx ) = ann ( x ) since S has the identity and using Lemma (1.2 bounded paper).

Therefore t ∈ ann S ( Η ) ⊆ ann S ( A ) .

Every cyclic bounded module is an Endo-G.B.

Let Η be a cyclic torsion-free S-module(where S is an integral domain), then Η is Endo-G.B.

From proposition (1.5 bounded paper), Η is bounded S-module and using Proposition 4.6 we conclude that Η is an End-G.B.

Let S be an integral domain and Η be a cyclic divisible multipilication S-module. Then Η is an End-G.B.

Every fully stable bounded S-module is an endo-GB.

Let Η be a fully stable bounded S-module, then by [coro 1.2 bounded] Η is cyclic and using(prop 3.7), the result is as follows.

Every finitely annihilated S-module is an End-GB S-module.

Because every bounded S-module is finitely annihilated and every End-GB S-module is bounded, the result is as follows.

Let Η be a multiplication End-G.B.S-module, then Η is finite generated of S-module.

Since every End-GB is finitely annihilated, then by [proposition 3.1 f.ann paper] finitely annihilated with multiplication gives the result.

Let Η be an End-G.B. fully stable semi-simple S-module, then Η is finitely generated.

Recall that an S-module Η is said that finendo if it is finitely generated over End ( Η ) . 12

Let Η be a semi-simple Endo-G.B.S-module, then Η is a finendo S-module.

Let Η be End-G.B, then Η is finite annihilated and by using [coro 2.3 f.ann paper] implies that Η is finendo.

Recall, an S-module Η is called a divisible if r Η = Η , for all 0 ≠ r ∈ S . 13

Let Η be a divisible Noetherian S-module, then Η is a finite annihilated if and only if Η is an Endo-G.B.

Suppose that Η is finite annihilated and let A ≤ Η . Then, N is a finitely generated submodule, and by the hypotheses, we have ann S ( Η ) = ann S ( A )

Now, let φ , ψ ∈ End ( H ) and as φ ( x ) = r x 1 + rx 2 + … + rx n , ∀ x ∈ Η , since Η is finitely generated, φ ( x ) ∈ Η and ψ ( x ) = x 1 + x 2 + … + x n , ∀ x ∈ Η , for some n ∈ Z + . Then φ ( x ) ∈ A so ann S ( A ) ⊆ ann S ( ( φoψ ) ( x ) ) .

Let t ∈ ann S ( ( φoψ ) ( x ) ) , then t ( φOψ ) ( x ) = 0 implies that t ( r x 1 + rx 2 + … + rx n ) = 0 so that tr ( x 1 + x 2 + … + x n ) = 0 . Hence tr ∈ ann S ( x 1 + x 2 + … + x n ) and tr ∈ ann S ( A ) = ann S ( Η ) . Therefore tr ( m ) = 0 , ∀ m ∈ Η . Thus, t ∈ ann S ( r Η ) = ann S ( Η ) ⊆ ann S ( A ) .

5. End-G.B. modules related to prime modules

In section concentrate on the relationship between Endo-G.B. and prime .

Every prime S-module is Bounded.

Let Η be a cyclic prime module, then Η is an End-G.B.

Since every prime module is bounded, then by Proposition (4.6) Η is an End-G.B.

Let Η be an Endo-G.B.S-module, 0 ≠ x ∈ Η such that the following conditions hold:

SX is an essentiol sub module of Η .

ann S ( x ) is prime ideal of S. Then Η is prime S-module.

Since Η is an End-G.B. module, then Η is bounded so that there exist x ∈ Η

ann S ( x ) = ann S ( Η ) . Let N ⊆ Η , there exist 0 ≠ t ∈ S , 0 ≠ tx ∈ N . It is obvious that ann S ( Η ) ⊆ ann S ( N ) and ann S ( N ) ⊆ ann S ( tx ) .

Let r ∈ ann S ( tx ) imply that r ( tx ) = 0 then rt ∈ ann S ( x ) and ann S ( x ) be prime ideals. Therefore, r ∈ ann S ( x ) = ann S ( Η ) . Therefore, ann S ( N ) ⊆ ann S ( tx ) ⊆ ann S ( x ) = ann S ( Η ) .

Hence, ann S ( N ) = ann S ( Η ) means that Η is prime modules.

Recall, a submodule A of an S-module Η is called pure if I Η ∩ A = IA , for every ideal I of S . 14

Let S be a principle ideal ring and Η is uniform F-regular S-module. Then Η is a prime module.

Let A ⊆ Η , then A be a pure submodule and r Η ∩ A = rA . It is clear that ann S ( Η ) ⊆ ann S ( A ) .

Now, let r ∈ ann S ( A ) , then rA = 0 . Thus, r Η ∩ A = 0 and since Η is uniform module, then A is an essential submodule. Therefore, r Η = 0 and r ∈ ann S ( Η ) .

Let Η be a cyclic F-regular uniform S-module, then Η is an End-G.B.module.

By previous proposition Η is prime module implies that Η is bounded module and since Η is a cyclic, then by (prop. 3.7) Η is an End-G.B.

There is a relationship between the scalar S-module and an End-GB, so we present some connections in the next properties and corollaries.

Remember that, let S be an integral domain and let Η be an S-module. An element h ∈ Η is called torsion if there is a nonzero element r ∈ S such that rh = 0 , denoted by T ( H ) . 15, 16

Let Η be a scalar torsion-free S-module, where(S is an integral domain) and every submodule is cyclic. Then Η is an End-G.B.

Let φ , ψ ∈ End ( Η ) and as φ ( x ) = rx and ψ ( x ) = x , ∀ x ∈ Η . Let N = SX be a sub- module of Η . Then φ ( x ) ∈ N so ann S ( N ) ⊆ ann S ( ( φO ψ ) ( x ) ) .

Let t ∈ ann S ( ( φO ψ ) ( x ) ) . Since Η is scalar torsion-free, then by coro[1.21 scalar thesis] Η is quasi-dedekind module.

Thus, t ( ( φO ψ ) ( x ) ) = 0 and t ( φ ( ψ ( x ) ) = o means that t ( φ ( x ) ) = 0 and φ ( tx ) = 0 hence tx = 0 since φ is monomorphism.

Because N is a cyclic submodule, then N is a bounded S-submodule. Therefore, t ∈ ann S ( N ) .

6. Conclusion

In this study, we presented a new class of S- modules called endo-greatly bounded modules and some results that explain its connection with the prime module under certain conditions, especially when we assume that an S-module H is a scalar module.

In addition, we compared some examples between our new module and bounded module, and we found that an Endo-G.B. The module is stronger than the bound module. Finally, we establish the behavior of the Endo-G.B. module with prime and it , s generalization like S-prime and strongly S-prime modules, by propositions and corollaries.

 Data availability

All data underlying the results are available as part of the article and no additional source data are required. Data sharing is not applicable to this paper as no datasets were generated or analysed during the current study.

 References Faith C : Algebra II Ring Theory. Berlin, Heidelberg: Springer Berlin Heidelberg;1976;191. 10.1007/978-3-642-65321-6 Mahmood L Al-Ani A : Bounded modules. Ibn AL-Haitham Journal For Pure and Applied Science. 2017;221(3):2838–2860. 10.1016/j.jpaa.2017.02.003 Al-Ani AS : FULLY BOUNDED MODULES. Journal of University of Anbar for Pure Science. Jun. 2012;1(1):65–70. 10.37652/juaps.2007.15520 Shihab BN : Scalar Reflexive Modules. PhD thesis. Iraq, Baghdad: College of Education Ibn AL-Haitham, University of Baghdad;2004. Shihab BN : On Almost Bounded Submodules. Ibn AL-Haitham Journal For Pure and Applied Science. 2010;23(2):168–174. Reference Source Murad MS Shihab BN : Endo-restricted bounded submodules related to prime submodules and scalar modules. AIP Conference Proceedings. AIP Publishing;2025, March;3264(1). 10.12988/imf.2014.47139 Jasem FD Elewi AA : 2-prime submodules of modules. Iraqi Journal of Science. Aug. 2022;3605–3611. 10.24996/ijs.2022.63.8.34 Beachy JA Blair WD : Finitely annihilated modules and orders in artinian rings. Communications in Algebra. Jan. 1978;6(1):1–34. 10.1080/00927877808822231 Naoum AG : Regular multiplication modules. Periodica Mathematica Hungarica. Oct. 1995;31(2):155–162. 10.1007/BF01876490 Haghany A Mazrooei M Vedadi MR : BOUNDED AND FULLY BOUNDED MODULES. Bulletin of the Australian Mathematical Society. Dec. 2011;84(3):433–440. 10.1017/S0004972711002814 Al-Mothafar NS Abdil-Khalik AJ : End-Ψ-Primary Submodules. International Journal of Science and Research (IJSR) ISSN. 2015;6. 10.21275/1061705 Monday CR : A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring. 2014. Reference Source Fiadh MS Shihab BN Abed MH : W-visible submodules and W-fully visible modules. Iraqi Journal for Computer Science and Mathematics. Jan. 2020;1–6. 10.52866/ijcsm.2019.01.01.005 Büyükaşık E Demir ÖI : Virtually regular modules. Journal of Algebra and Its Applications. Jan. 2025. 10.1142/S0219498826501021 Ashby WT : On graded torsion free and graded injective k[x] modules. International Journal of Algebra. 2023;17(1):13–20. 10.12988/ija.2023.91735 Hadi IMA Hamil MA : Torsion (Torsion Free) Fuzzy Modules Over Fuzzy Integral Domain. Ibn AL-Haitham Journal For Pure and Applied Science. 2017;20(1):160–171. 10.5256/f1000research.189933.r454437 Reviewer response for version 1 Aydoğdu Pınar 1 Referee https://orcid.org/0000-0002-2148-2980 Hacettepe University, Beytepe, Turkey

Competing interests: No competing interests were disclosed.

 6 2 2026 Copyright: © 2026 Aydoğdu P 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

The manuscript introduces a new class of $S$-modules termed "Endo-Greatly Bounded (Endo-G.B.) modules". The authors attempt to establish connections between this new class and established concepts such as bounded and Endo-R.B. modules. While the topic is relevant to module theory, the current manuscript lacks the mathematical depth, formal rigor, and motivation required for a research article.

​​​​​​The manuscript fails to provide a clear motivation for introducing "Endo-G.B. modules". It is not explained what specific mathematical problem this new class solves or how it enhances our understanding of module structures beyond existing definitions like "Bounded" or "Endo-R.B." modules. Without a clear justification or an application that necessitates this specific property, the study appears to be a formal exercise in generalizing definitions rather than a contribution to the field.

The primary definition (Definition 2.1) is recursive: "An $S$-module $H$ is called Endo-Greatly Bounded module if every proper submodule of $H$ is an Endo-Greatly Bounded module". This lacks a structural base case or a descriptive property. Consequently, most subsequent proofs become repetitive applications of this recursive logic rather than insightful mathematical derivations.

The results (e.g., Propositions 3.5, 3.11, 3.13) are elementary and follow directly from the standard properties of modules (quotients, direct sums, and homomorphic images). These results resemble textbook exercises and do not offer significant new insights into the theory.

The manuscript contains numerous typographical errors and non-standard notation that hinder professional readability. For instance, the annihilator is inconsistently abbreviated (e.g., "amn", "tamn", "rnn"), and standard terms are misspelled (e.g., "prim module", "finitaly", "fulley").

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

No

Reviewer Expertise:

Ring and Module Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.