F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172196.1 Research Article Articles New Class of S-Pseudo Bounded Modules With Some Related Concepts

[version 1; peer review: 1 not approved]

 Madhi Rashid Amal Writing – Review & Editing https://orcid.org/0009-0005-6526-4140 a 1 2 Najad Shihab Buthyna Writing – Original Draft Preparation 1 2 Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 31001, Iraq Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 10001, Iraq amal.madhi2203@ihcoedu.uobaghdad.edu.iq

No competing interests were disclosed.

 9 12 2025 2025 14 1385 2 12 2025 Copyright: © 2025 Madhi Rashid A 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.

In the present study , every module M is unitary and every ring F is commutative with identity. We gave a definition of a new class F − module which is namely S-pseudo bounded module symbolically (S-PS.B. F − module) and introduced some different approaches to attach this class with other types of well-known modules such that monoform module, quasi-Dedekind module, compressible module and retractable module. The main purpose of this article is to present a few new conditions for some corollaries and properties. The F − homomorphism of monoform and compressible modules connect in a useful way with an endomorphism of a F − module M that we relied on it in the definition of S-pseudo bounded module. We used the symbol End ( M ) which means the set of all endomorphism maps of F − module M . Also S-pseudo bounded module gave us directly or with some conditions different modules such as retractable module, an injective module and others.

 S-Pseudo bounded module Fully polyform module Multiplication module Critically compressible module Scalar module. The author(s) declared that no grants were involved in supporting this work. 1. Introduction

The notion of bounded module was studied by Carl Faith where (if there exists x ∈ M such that ann F ( M ) = ann F ( x ) then M is said to be bounded). 1 Moreover, the concept of bounded submodule introduced with details by AL-LNI where (if there exists x ∈ ℵ such that ann F ( ℵ ) = ann F ( x ) , then ℵ is said to be bounded submodule of M ). 2 The concept of almost bounded submodule was submitted by Buthyna Najad where (if there exists an element x ∈ M , x ∉ ℵ such that ann F ( ℵ ) = ann F ( x ) then ℵ is called almost bounded submodule). 3 Also, scalar modules and prime modules are involved in several properties as a condition to attach S-Pseudo bounded module with other modules. Note that if ann F ( ℵ ) = ann F ( x ) , for every submodule ℵ of M then M is said to be prime module. 4 Furthermore, if M is finitely generated , then M is compressible F − module if and only if it is uniform and prime. 5 This study investigates how S-PS.B. modules relate to other well-known module types offering new insights into these connections. We explore how these modules relate to monoform modules, compressible modules, critically compressible modules and Rickart modules directly or with some conditions.

In this article, we investigate a new class of module called S-Pseudo bounded module where in Section 2 some related facts are reviewed and Section 3 contained some definitions with examples while in Section 4 we introduced some significant relationships and connected with other modules.

 2. Preliminaries

1 A F − module M is bounded module if there exists an element x ∈ M such that ann F ( M ) = ann F ( x ) .

6 A F − module M is said to be scalar if for every φ ∈ End ( M ) there exists r ∈ F such that φ ( x ) = rx , ∀ x ∈ M .

7 A F − module M is called monoform if ∀ ℵ ≠ 0 of M is dense where a submodule ℵ of M is dense if for any x , y ∈ M , x ≠ 0 there exists t ∈ F such that ty ∈ ℵ and tx ≠ 0 .

Equivalently, A F − module M is said to be monoform if ∀ ℵ ≠ 0 of M and ∀ 0 ≠ φ ∈ Hom ( ℵ , M ) then φ is monomorphism. 7 Also , if M is monoform module, then it is uniform and prime and hence we deduce that ann F ( M ) is prime ideal of F .

8 A F − module M is said to be finitely annihilated F − module if there exists a finitely generated submodule ℵ of M such that ann F ( M ) = ann F ( ℵ ) .

9 A F − module M is called quasi-Dedekind if ∀ ℵ ≠ 0 of M is quasi-invertible where ℵ is quasi-invertible if Hom ( M / ℵ , M ) = 0 .

Equivalently, a F − module M is said to be quasi-Dedekind if for each non-zero endomorphism of M is a F − monomorphism .

10 A F − module M is called polyform if every essential submodule ℵ of M is dense. Note that every monoform is polyform.

10 A F − module M is said to be fully polyform if every P-essential submodule of M is dense where ℵ is called P-essential if every pure submodule k of M such that ℵ ∩ k = ( 0 ) implies that k = ( 0 ) .

10 A F − module M is said to be fully retractable if for every non-zero submodule ℵ of M and each non-zero homomorphism f ∈ Hom ( ℵ , M ) implies that Hom ( ℵ , M ) ≠ 0 .

11 A F − module M is called coprime if ann F ( M ) = ann F ( M / ℵ ) for every proper submodule ℵ of M .

6 If M is a multiplication finitely generated F − module , then M is a scalar F − module .

6 Let M be an injective scalar F − module . Then ℵ is a scalar submodule of M .

12 Every finitely generated F − module is finitely annihilated.

12 Let M be a multiplication F − module . Then M is finitely generated if and only if M is finitely annihilated.

6 Let M be a scalar torsion-free F − module with ( F is an integral domain). Then every φ ∈ End ( M ) is F − monomorphism .

13 Let M be a quasi-Dedekind retractable F − module . If every 0 ≠ φ ∈ End ( M ) is a monomorphism, then M is compressible module.

14 Let M be a retractable F − module such that End( M ) is a domain. Then M is critically compressible module if and only if it is polyform.

14 Let M be a retractable F − module . Then M is critically compressible module if and only if every non-zero partial endomorphism of M is monomorphism.

14 Let M be a fully retractable F − module with End( M ) is a domain. Then M is polyform.

15 A F − module M is called Rickart F − module if and only if Kerφ is a direct summand of M .

15 If M is an injective prime F − module , then M is Rickart module.

16 A F − module M is called N-Rickart if for every homomorphism g : M → ℵ , ker g is a summand of M .

14 If M is uniform module, then M is fully polyform if and only if M is monoform module.

17 If M is finitely generated F − module , then M is compressible F − module if and only if M is uniform prime module.

3. S-Pseudo bounded modules

In this part, a new class of an F − module will be investigated with some definitions and examples related to S-PS.B. F -module which depends on an endomorphism map over an F − module M .

A proper submodule ℵ of a F − module M is called S-pseudo bounded F − submodule (symbolically S-PS.B. F -submodule) if there exists φ ( x ) ∈ S = End ( M ) such that φ ( x ) ∈ ℵ for some x ∈ M implies that ann F ( ℵ ) = ann F ( φ ( x ) ) .

1- Let M = ℤ 2 ⨁ ℤ 4 as a ℤ 8 − module and ℵ = ℤ 2 ⨁ ⟨ 2 ¯ ⟩ . Define φ : M ⟶ M by φ ( a ¯ , b ¯ ) = ( a ¯ , 2 ¯ ) , ∀ ( a ¯ , b ¯ ) ∈ M . If ( 0 ¯ , 2 ¯ ) ∈ M then φ ( 0 ¯ , 2 ¯ ) = ( 0 ¯ , 2 ¯ ) ∈ ℵ . Therefore, ann ℤ 8 ( ℵ ) = ann ℤ 8 φ ( 0 ¯ , 2 ¯ ) , and hence ℵ is S-PS.B. ℤ 8 − submodule of M .

2- Consider M = ℤ ⨁ ℤ 2 as a ℤ − module and ℵ = 2 ℤ ⨁ ⟨ 0 ¯ ⟩ , then there exists φ : M ⟶ M defined by φ ( a , b ¯ ) = ( a , 0 ¯ ) , ∀ ( a , b ¯ ) ∈ M clearly, φ ∈ End ( M ) . Now, suppose that x = ( 2 , 1 ¯ ) ∈ M . Sequently, φ ( x ) = φ ( 2 , 1 ¯ ) = ( 2 , 0 ¯ ) ∈ ℵ . Thus ann ℤ ( ℵ ) = ann ℤ φ ( 2 , 1 ¯ ) = ⟨ 0 ¯ ⟩ . Therefore, ℵ is S-PS.B. ℤ − submodule of M .

3- Suppose that ℤ 6 as a ℤ − module and M = ℤ 6 , ℵ = ⟨ 3 ¯ ⟩ . An endomorphism φ : M ⟶ M defined by φ ( x ¯ ) = 0 ¯ , ∀ x ¯ ∈ ℤ 6 , φ ( x ¯ ) ∈ ℵ if x ¯ = 2 ¯ ∈ ℤ 6 , we have ℤ = ann ℤ φ ( 2 ¯ ) = ann ℤ ( 0 ¯ ) ≠ ann ℤ ⟨ 3 ¯ ⟩ = 2 ℤ , ℵ is not S-PS.B. ℤ − submodule.

We establish that every Endo-R.B. is S.PS.B. submodule but the converse is not necessary true in general. If ℵ is an Endo-R.B. then there exists φ ∈ End ( M ) such that φ ( x ) ∈ ℵ , for some x ∈ M implies that ann F ( ℵ ) = ann F φ ( x ) by Ref. 7. But ann F φ ( x ) = { r ∈ F : r n ∈ ann F φ ( x ) , n ∈ ℤ + } and from above equality we get ann F φ ( x ) = ann F ( ℵ ) . Hence, ℵ is S-PS.B. F − submodule. But the converse is not true in general for example:

Let M = ℤ 2 ⨁ ℤ 4 as ℤ − module . Define φ : M ⟶ M as: φ ( a ¯ , b ¯ ) = ( 0 ¯ , b ¯ ) , ∀ ( a ¯ , b ¯ ) ∈ M if we take ℵ = ⟨ 0 ¯ ⟩ ⨁ ℤ 4 , ( 1 ¯ , 2 ¯ ) ∈ M then φ ( 1 ¯ , 2 ¯ ) = ( 0 ¯ , 2 ¯ ) ∈ ℵ , implies that ann ℤ φ ( 1 ¯ , 2 ¯ ) = ann ℤ ( 0 ¯ , 2 ¯ ) = 2 ℤ = 2 ℤ and ann ℤ ( ⟨ 0 ¯ ⟩ ⨁ ℤ 4 ) = 4 ℤ = 2 ℤ .Thus ℵ is S-PS.B. ℤ − submodule, but ann ℤ ( ⟨ 0 ¯ ⟩ ⨁ ℤ 4 ) = 4 ℤ is not equal to ann ℤ φ ( 1 ¯ , 2 ¯ ) = ann ℤ ( 0 ¯ , 2 ¯ ) = 2 ℤ . Therefore ℵ is not Endo-R.B. ℤ − submodule.

A F − module M is said to be S-PS.B. F − module if every proper submodule of M is S-PS.B F − submodule.

ℤ p as a ℤ − module is S-PS.B. ℤ − module , where p is prime, since the only proper submodule of ℤ p is ⟨ 0 ¯ ⟩ . If we define φ : ℤ 2 ⟶ ℤ 2 , φ ∈ End ( ℤ 2 ) as φ ( x ¯ ) = 2 x ¯ , ∀ x ¯ ∈ ℤ 2 , then φ ( x ¯ ) ∈ ⟨ 0 ¯ ⟩ and ann ℤ ⟨ 0 ¯ ⟩ = ann ℤ φ ( x ¯ ) = ℤ . Hence ℤ p is S-PS.B. ℤ − module .

Consider ℤ 4 as a ℤ − module . Define φ : ℤ 4 ⟶ ℤ 4 as φ ( a ¯ ) = 0 ¯ , ∀ a ¯ ∈ ℤ 4 . If we take M = ℤ 4 , ℵ = ⟨ 2 ¯ ⟩ , then φ ( a ¯ ) ∈ ℵ , hence ℤ 4 is not S-PS.B. ℤ − module , since if a ¯ = 3 ¯ ∈ ℤ 4 , then 2 ℤ = ann ℤ ( ℵ ) ≠ ann ℤ φ ( 3 ¯ ) = ℤ .

4. S-Pseudo bounded modules with some modules

In this section, many modules played a major role in getting S-Pseudo bounded module such that monoform, compressible, quasi-Dedekind module and others modules. We get more results through several relationships.

If M is a multiplication torsion-free F − module with ( F is an integral domain), then the following statements are equivalent

( i ) M is S-PS.B. F − module .

( ii ) M is monoform module .

( i ) ⟹ ( ii ) Suppose that M is S-PS.B. F − module . Since M is a multiplication torsion-free, then M is finitely generated, by Remark (2.12) and Proposition (2.13). Therefore M is scalar by Corollary (2.10). By Proposition (2.14), every φ ∈ End ( M ) is monomorphism. Suppose that φ : M ⟶ M be an F − homomorphism and i : ℵ ⟶ M is inclusion map, then φ ∘ i : ℵ ⟶ M is monomorphism and M is monoform module.

( ii ) ⟹ ( i ) Let M is monoform module implies that M is uniform prime module and every submodule ℵ ≠ 0 of M is dense. Suppose that φ ∈ End ( M ) such that φ : M ⟶ M and φ ( x ) = tx , 0 ≠ x ∈ M , 0 ≠ t ∈ F . Since M is uniform, then tx ∈ ℵ . Hence ann F ( ℵ ) ⊆ ann F φ ( x ) and thus ann F ( ℵ ) ⊆ ann F φ ( x ) . Let a ∈ ann F φ ( x ) implies that a n . φ ( x ) = 0 , for some n ∈ ℤ + which implies a n . ( tx ) = 0 and a n t ( x ) = 0 . Since ℵ is dense submodule and ann F ( M ) is prime ideal of F , then t ∉ ann F ( x ) and a n ∈ ann F ( x ) = ann F ( M ) ⊆ ann F ( ℵ ) . Thus ann F ( ℵ ) = ann F φ ( x ) and M is S-PS.B. F − module .

However, the condition torsion-free is suffice to prove M is monoform module.

Let M be a torsion-free S-PS.B. F − module . Then M is monoform module.

Suppose that ℵ be any non-zero submodule of M . Since M is a torsion-free module, then ∀ x ∈ M there exists 0 ≠ t ∈ F such that tx = 0 , we obtain x = 0 . Since M is S-PS.B. F − module , then ∃ φ ∈ End ( M ) such that φ : M ⟶ M and define as φ ( x ) = tx , x ∈ M and hence tx ∈ ℵ . Now, assume that 0 ≠ x ∈ M , we have to show that tx ≠ 0 . Let tx = 0 implies that x = 0 , since M is torsion-free module which is contradiction and thus tx ≠ 0 . Since ℵ is an artibrary submodule, then M is monoform module.

If M is quasi-Dedekind S-PS.B. F − module , then M is monoform module.

It is sufficient to show that every non-zero submodule of M is dense. Assume that ℵ is any non-zero submodule of M . Define φ : M ⟶ M by φ ( x ) = tx , ∀ x ∈ M . Suppose that 0 ≠ x ∈ M and tx = 0 and since M is S-PS.B. F − module implies that tx ∈ ℵ and φ ( x ) = 0 which means x ∈ ker φ . Since M is quasi-Dedekind module, then φ is monomorphism. Hence x = 0 which is contradiction. Thus tx ≠ 0 and ℵ is dense submodule.

Every S-PS.B. F − module is retractable F − module .

Assume that ℵ is a non-zero submodule of S-PS.B. F − module M , then there exists 0 ≠ φ ∈ End ( M ) such that φ : M ⟶ M and defined as φ ( x ) = tx , 0 ≠ x ∈ M , φ ( x ) ∈ ℵ and ann F ( ℵ ) = ann F φ ( x ) . Let Hom ( M , ℵ ) = 0 and i : ℵ ⟶ M the inclusion map. Then φ = i ∘ g where g : M ⟶ ℵ , g = 0 . Thus φ ( x ) = ( i ∘ g ) ( x ) = i ( g ( x ) ) = 0 that means ann F ( ℵ ) ≠ ann F φ ( x ) which is a contradiction. Hence Hom ( M , ℵ ) ≠ 0 ∀ ℵ ≠ 0 of M . Therefore M is retractable F − module .

Conversely is not true in general, we have this example:

Consider ℤ 4 as a ℤ − module and M = ℤ 4 , ℵ = ⟨ 2 ¯ ⟩ . Define φ : ℤ 4 ⟶ ℤ 4 as φ ( x ¯ ) = 0 ¯ , ∀ x ¯ ∈ ℤ 4 , φ ∈ End ( ℤ 4 ) . Since φ ( x ¯ ) ∈ ℵ , ∀ x ¯ ∈ ℤ 4 , Im φ ⊆ ℵ then M is retractable module. But M is not S-PS.B ℤ − module , since 2 ℤ = ann ℤ ( 2 ¯ ) ≠ ann ℤ φ ( 1 ¯ ) = ℤ , where 1 ¯ ∈ ℤ 4 .

If M is S-PS.B. torsion-free module, then M is critically compressible module.

By Corollary (4.2) and Proposition (2.17) we get M is critically compressible.

Let M be S-PS.B. F − module , then M is critically compressible F − module if and only if every non-zero partial endomorphism of M is monomorphism.

If M is a duo S-PS.B. F − module , then M is fully retractable module.

Since M is S-PS.B. F − module , then there exists φ is a non-zero endomorphism of M such that φ ( x ) ∈ ℵ , x ∈ M and ann F ( ℵ ) = ann F φ ( x ) where ℵ is a submodule of M . For every φ ∈ End ( M ) we have φ ( ℵ ) ⊆ ℵ , since M is a duo. Thus the partial endomorphism of M is not zero and 0 ≠ φ : ℵ ⟶ M . Therefore M is retractable module, by Proposition (4.4) so that there exists 0 ≠ ψ : M ⟶ ℵ a homomorphism. Hence ψ ∘ φ ≠ 0 and thus M is fully retractable module.

If M is a duo S-PS.B. F − module and End ( M ) is a domain, then M is polyform module.

Assume that M is S-PS.B. F − module then by previous proposition, M is fully retractable module. Applying Proposition (2.18) we get the result.

Let M be an uniform S-PS.B. F − module and End ( M ) is a domain. Then the following statements are equivalent:

( i ) M is critically compressible module.

( ii ) M is polyform module.

( i ) ⟹ ( ii ) Suppose that M is critically compressible module, then M is monoform, by Corollary(4.6) and thus M is polyform.

( ii ) ⟹ ( i ) Assume that M is polyform module, then Corollary (2.22), M is monoform, since M is uniform. Since M is S-PS.B. F − module , then M is retractable. By Proposition (2.16), M is critically compressible module.

If M is S-PS.B. uniform F − module where End ( M ) is a domain, then the following statements are equivalent:

( i ) M is fully polyform module.

( ii ) M is critically compressible module.

If M is uniform, then polyform and fully polyform are equivalent, by Ref. 10.

If M is a compressible F − module , then M is S-PS.B. F − module .

Assume that φ ∈ End ( M ) where φ : M ⟶ M define as φ ( x ) = tx , x ∈ M . For each submodule ℵ ≠ 0 of M there exists h : M ⟶ ℵ a monomorphism map, since M is a compressible module. If φ = i ∘ h where i is the inclusion map, then φ ( x ) = ( i ∘ h ) ( x ) = i ( h ( x ) ) = h ( x ) ∈ ℵ . Suppose that a ∈ ann F φ ( x ) implies that a n . φ ( x ) = 0 , for some n ∈ ℤ + , ∀ x ∈ M . Thus a n . h ( x ) = 0 , ∀ x ∈ M so that h ( a n x ) = 0 and h ( a n x ) = h ( 0 ) which means a n x = 0 , ∀ x ∈ M . Hence a ∈ ann F ( M ) and a n ∈ ann F ( M ) = ann F ( ℵ ) , since M is prime module. Therefore ann F φ ( x ) = ann F ( ℵ ) and M is S-PS.B. F − module .

Let M be a torsion-free multiplication F − module with ( F is an integral domain).Then M is S-PS.B. F − module if and only if M is a compressible module.

Let M be S-PS.B. F − module . By Corollary (2.10) M is a scalar module and hence every φ ∈ End ( M ) is monomorphism. Since M is retractable module by Proposition (4.4), then there exists a homomorphism 0 ≠ f : M ⟶ ℵ for every submodule ℵ ≠ 0 of M . Hence φ = i ∘ f is monomorphism where i is an inclusion map. Thus f is a monomorphism and hence M is a compressible module.

Conversely, applying previous proposition.

If M is a quasi-Dedekind retractable F − module , then M is S-PS.B. F − module .

Every F − homomorphism φ ∈ End ( M ) is monomorphism, since M is a quasi-Dedekind. Hence M is a compressible module, by Proposition (2.15) and thus M is S-PS.B. F − module , by Proposition (4.11).

If M is a finitely generated, then every uniform prime module is S-PS.B. F − module .

Since M is a finitely generated implies that M is a compressible module, by (2.23). The result is obtained by Proposition (4.11).

If M is a quasi-Dedekind F − module and φ ( M ) ⊈ ∩ ψ ∈ Hom ( F , ℵ ) kerψ where φ ∈ Hom ( M , F ) and ℵ is a submodule of M , then M is S-PS.B. module .

Suppose that φ ( M ) ⊈ ∩ ψ ∈ Hom ( F , ℵ ) kerψ , then ∃ ψ ∗ : F → ℵ such that 0 ≠ ψ ∗ ∘ φ ∈ Hom ( M , ℵ ) . Thus M is retractable module. Since M is a quasi-Dedekind module, then M is S-PS.B. F − module by proposition (4.13).

Let M is a quasi-Dedekind S-PS.B. F − module . Then ann F ( M ) ≠ ann F ( M ℵ ) , ℵ ≤ M .

Assume that ℵ ≤ M and ann F ( M ) = ann F ( M ℵ ) . Hence [ ℵ : F M ] = ann F ( M ) = ann F ( x ) , ∀ x ∈ M , since M is prime. Thus for every t such that t M ⊆ ℵ and ty ∈ ℵ so that tx = 0 , x ≠ 0 , x , y ∈ M which is contradiction since M is monoform module by Proposition (2.18). Thus ann F ( M ) ≠ ann F ( M ℵ ) .

Let M is a quasi-Dedekind S-PS.B. F − module . Then M is not coprime F − module .

Let ℵ ≤ M and M / ℵ be a quasi-Dedekind F − module . Then M is S-PS.B. F − module .

Suppose that φ ∈ End ( M ) and φ : M ⟶ M defined as φ ( n ) = tn , n ∈ M . Since M / ℵ be a quasi-Dedekind module so if we define ψ : M / ℵ → M / ℵ as ψ ( x + ℵ ) = tx + ℵ , ∀ x ∈ M then either ψ = 0 or ψ is a monomrphism. Let ψ ≠ 0 and n + ℵ ∈ Ker ψ then ψ ( n + ℵ ) = ℵ which means tn + ℵ = ℵ . Therefore, n + ℵ = ℵ so that n ∈ ℵ and tn ∈ ℵ , ∀ n ∈ ℵ . Hence ann F ( ℵ ) ⊆ ann F φ ( n ) . If ψ = 0 we obtain tn + ℵ = ℵ so tn ∈ ℵ and also ann F ( ℵ ) ⊆ ann F φ ( n ) . Now, let a ∈ ann F φ ( n ) so a n . φ ( n ) = 0 , for some n ∈ ℤ + and a n ( tn ) = 0 then a ∈ ann F ( tn ) , ∀ tn ∈ ℵ . Thus a ∈ ann F ( ℵ ) and we get ann F ( ℵ ) = ann F φ ( n ) .

If M is S-PS.B. F − module , then M ≠ t M .

Assume that M = t M and ∃ φ ∈ End ( M ) such that φ : M ⟶ M and defined as φ ( x ) = tx , ∀ x ∈ M , since M is S-PS.B. F − module . Thus M is retractable by (4.4) which means Imφ ⊆ ℵ for every submodule ℵ ≠ 0 of M . Therefore φ ( M ) ⊆ ℵ so that t M ⊆ ℵ and since M = t M implies M ⊆ ℵ which is contradiction. Thus M ≠ t M .

Let M is S-PS.B. F − module such that 0 ≠ φ ∈ End ( M ) . Then φ is not epimorphism.

Suppose that φ ∈ End ( M ) . φ ≠ 0 Since M is S-PS.B. F − module , then M is retractable module ad there exists a homomorphism 0 ≠ f : M ⟶ ℵ , for every submodule ℵ ≠ 0 of M . Put φ = i ∘ f : M ⟶ M where i : ℵ ⟶ M is the inclusion map. Therefore φ is not epimorphism.

Let M be S-PS.B. torsion-free F − module . Then M is an injective.

Since M a torsion-free and S-PS.B. F − module , so that M is a monoform module by Corollary (4.2) and thus there exists a non-zero monmorphism g : ℵ ⟶ M defined by g ( y ) = y , ∀ y ∈ ℵ . Since M is S-PS.B. F − module , then M is retractable and there exists a homomorphism 0 ≠ f : M ⟶ ℵ defined by f ( y ) = y , ∀ y ∈ M . Moreover we consider the diagram

Suppose that y ∈ M , then ( g ∘ f ) ( y ) = g ( f ( y ) ) = g ( y ) = I M . Thus g ∘ f = I M and the diagram is commutative so M is an injective F − module .

If M is a torsion-free and S-PS.B. F − module , then End ( M ) has no zero-divisors.

Suppose that φ , ψ ∈ End ( M ) are two non-zero homomorphism implies that ∃ n , n ′ ∈ M such that φ ( n ) = x ≠ 0 , ψ ( n ′ ) = y ≠ 0 where x , y ∈ M . Since M is monoform by (4.2) and thus M is quasi-Dedekind module. Hence φ , ψ ∈ are two F − monomorphisms. Thus ( φ ∘ ψ ) ( n ′ ) = φ ( y ) ≠ 0 and ( ψ ∘ φ ) ( n ) = ψ ( x ) ≠ 0 which means End ( M ) has no zero-divisors.

If M is torsion-free S-PS.B., then M is a Rickart F − module .

Since M is an injective, by Proposition (4.21) and hence M is prime module since every torsion-free S-PS.B. F − module is monoform. By Proposition (2.20) we obtain that M is a Rickart.

Let M be an indecomposable S-PS.B. F − module with M is N-Rickart. Then M is quasi-Dedekind.

Suppose that M is S-PS.B. F − module , then M is a retractable module, by (4.4) and thus Hom ( M , ℵ ) ≠ 0 for every non-zero submodule ℵ of M . Suppose that φ : M → M is an endomorphism of M . Since M is N-Rickart, then kerφ is direct summand of M . Since M be an indecomposable so that kerφ = 0 . Thus φ is a F − monomorphism and M is quasi-Dedekind module.

Let M be a torsion-free S-PS.B. Then for each φ ∈ End ( M ) there exists once φ is splits in M .

From (4.23) we get M is a Rickart F − module , Consider the following short exact sequence 0 → kerφ = ann M φ ( x ) → M → φ M → 0 .

Consequently, kerφ is direct summand of M , since M is Rickart module. But kerφ = ann M φ ( x ) which means φ is splits in M .

Let M be a S-PS.B. F − module . Then End ( M ) is a Rickart ring if and only if M is a Rickart module.

Since M is a S-PS.B. F − module , which means M is retractable module so that we get the result by (proposition (3.3), Ref. 16).

5. Conclusion

In our article, we have a new class of module called S-Pseudo Bounded and explain the relation of this module with other modules. Also, we introduced many nicely properties that join S-Pseudo Bounded with important modules such that monoform modules, retractable modules, quasi-Dedekind modules, and compressible modules. In addition, using S-Pseudo Bounded F − module as supposition lead us to get some statements that will be important for other who want to study in this field.

 Ethical considerations

This research did not involve any studies with human participants or animals and therefore did not require ethical approval.

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No experimental data were generated or analyzed in this study. The research is entirely theoretical within the field of pure mathematics (abstract algebra); therefore, data sharing is not applicable. No datasets were generated or analyzed during the current study. All results are theoretical and derived analytically within the framework of abstract algebra.Therefore, data sharing is not applicable to this article as no datasets were created or used.

 References Faith C : Algebra II Ring Theory: Vol. 2: Ring Theory. Vol. 191. Springer Science & Business Media;2012. 10.1007/978-3-642-65321-6 Mahmood LS Al-Ani AS : Bounded Modules. Ibn Al-Haitham J. Pure Appl. Sci. 2017;19(3):75–91. Reference Source Shihab BN : On Almost Bounded Submodules. IIbn Al-Haitham J. Pure Appl. Sci. 2017;23(2):168–174. Reference Source Ramadhan HA Al Mothafar NS : Weakly Small Semiprime Submodules. J. Phys. Conf. Ser. 2021;1879(3):032128. IOP Publishing. 10.1088/1742-6596/1879/3/032128/, Hadi IMA Al-Aeashi SN Shyaa FD : T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules. Baghdad Sci. J. 2021;18(1):0156. 10.21123/bsj.2021.18.1.0156 Shihab BN : Scalar Reflexive Modules. (Doctoral dissertation, Ph.D. Thesis). Iraq: University of Baghdad;2004. Reference Source Murad MS Shihab BN : Endo-restricted bounded submodules related to prime submodules and scalar modules. AIP Conf. Proc. 2025;3264(1). AIP Publishing LLC. 10.1063/5.0258500 Abdul-Al-Kalik AJ : Semi–Bounded Modules. Baghdad Sci. J. 2012;9(4):20. 10.21123/bsj.2012.9.4.720-727 Al-Quraishi AD Shihab BN : Pu-visible Submodule with their most prominent characteristics. IOP Conf. Ser.: Mater. Sci. Eng. 2020;871(1). IOP Publishing. Reference Source Muhammad M Ahmed MA : Fully polyform modules and related concepts. Iraqi J. Sci. 2024;3313–3330. 10.24996/ijs.2024.65.6.30 Ali IM Ghawi T'a Y : Some results on essentially quasi-Dedekind modules. Iraqi J. Sci. 2015;56(2A):1130–1139. Reference Source Mijbass AS Mohammadali HK : On Finitely Annihilated Modules. Kirkuk J. Sci. 2008;3(2):117–133. 10.32894/kujss.2008.42471 Hakeem A Hashim MB Al-Mothafar NS : Closed (st-closed) compressible modules and closed (st-closed) retractable modules. Iraqi J. Sci. 2024;2141–2149. 10.24996/ijs.2024.65.4.31 Reference Source Rodrigues VS Alves A Sant’Ana. : A note on a problem due to Zelmanowitz. Algebra and Discrete Mathematics. 2009. Reference Source Rahman MQ Al-Bahrani BH : On Rickart modules. Iraqi J. Sci. 2019;2473–2477. 10.24996/ijs.2019.60.11.18 Lee G Tariq Rizvi S Roman CS : Rickart modules. Commun. Algebra. 2010;38(11):4005–4027. 10.1080/00927872.2010.507232 Al-Mothafar NS Abdil-Khalik AJ : End-y-Prime Submodules. Ibn Al-Haitham J. Pure Appl. Sci. 2016;29(2). Reference Source 10.5256/f1000research.189909.r464812 Reviewer response for version 1 Kyomuhangi Annet 1 Referee Busitema University, Tororo, Uganda

Competing interests: No competing interests were disclosed.

 2 4 2026 Copyright: © 2026 Kyomuhangi A 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

Authors introduce a new class of modules called S-Pseudo Bounded modules and give enough examples. They also relate this notion to existing notions such as Rickart modules, Compressible modules,Dedekind modules, Endo R.B modules, monoform and polyform modules which have been studied extensively indicating that it is an active research area.  However, for a paper to read well, I suggest that you replace A F-module with 'An' F-module.

In Section 3,

2. I would have loved to see more properties of  S.PS.B. modules. For instance, closure properties; under submodules, quotients, direct sums, products, summands and extensions.

3. Relate  S.PS.B. modules with other notions such as bounded and almost bounded modules

4. Some terms such as Compressible, critically compressible modules and a partial endomorphism need to be defined. You defined N-Rickart modules but did not specify what a rickart module is. 

5.  The research extends the study of Endo-R.B modules as it is shown that every Endo-R.B module is  S.PS.B. With this result, most of the results in  Section 4 are a consequence of what was studied in   A New Class of Endo-R.B Module and Its Relationship with Modules.   For instance, Propositions 4.1-4.20 in this paper are consequences of Propositions 4.5-5.8 in   A New Class of Endo-R.B Module and Its Relationship with Modules. 

6. In Propositions 4.18 and 4.19, you use 't' in your statements but do not indicate what it is. Could it be an element of F?

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?

Yes

Reviewer Expertise:

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.

Rashid Amal Mathematics, College of Education for Pure Sciences, Ibn-AL-Haitham, University of Baghdad, Heet, Anbar, Iraq

Competing interests: No competing interests

 2 4 2026

Thank you for your valuable comments and constructive feedback. We sincerely appreciate your time and effort in reviewing our manuscript. All comments have been carefully considered, and the paper has been revised accordingly, incorporating your suggestions as much as possible. We will submit the revised version based on these modifications.

1- We replaced A F-module with 'An' F-module.

2- We added more  properties of  S.PS.B. modules at the end of Section 3.

3- We can relate S.PS.B. modules with other notions such as bounded and almost bounded but it requires many conditions to reach what we want. For instance, for bounded we need every ideal I of F is semiprime.

4- We added the definitions of Compressible, critically compressible module and a partial endomorphism in the end of section 2. The definition of Rickart module already exists in the section 2 as 2.19

5- In Propositions 4.18 and 4.19, 't' be an element of F.