New Class of S-Pseudo Bounded Modules With Some Related Concepts

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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." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/14-1385/v3", "name": "New Class of S-Pseudo Bounded Modules With Some Related Concepts" } } ] } Home Browse New Class of S-Pseudo Bounded Modules With Some Related Concepts ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Madhi Rashid A and Najad Shihab B. New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.12688/f1000research.172196.3 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Revised New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] Amal Madhi Rashid https://orcid.org/0009-0005-6526-4140 1,2 , Buthyna Najad Shihab 1,2 Amal Madhi Rashid https://orcid.org/0009-0005-6526-4140 1,2 , Buthyna Najad Shihab 1,2 PUBLISHED 16 May 2026 Author details Author details 1 Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 31001, Iraq 2 Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 10001, Iraq Amal Madhi Rashid Roles: Writing – Review & Editing Buthyna Najad Shihab Roles: Writing – Original Draft Preparation OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract 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. READ ALL READ LESS Keywords S-Pseudo bounded module, Fully polyform module, Multiplication module, Critically compressible module, Scalar module. Corresponding Author(s) Amal Madhi Rashid ( [email protected] ) Close Corresponding author: Amal Madhi Rashid Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 Madhi Rashid A and Najad Shihab B. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Madhi Rashid A and Najad Shihab B. New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.12688/f1000research.172196.3 ) First published: 09 Dec 2025, 14 :1385 ( https://doi.org/10.12688/f1000research.172196.1 ) Latest published: 16 May 2026, 14 :1385 ( https://doi.org/10.12688/f1000research.172196.3 ) Revised Amendments from Version 2 1- we mentioned some more recent works in the introduction part based on modules in logical algebras such as modules over JU algebras and some other structures and consequently add in the reference part. 2- We Shift the definition of bounded modules from introduction part to Preliminaries part. Definitions 2.1, 2.27 3- We Revise the English once again throughout the paper. 1- we mentioned some more recent works in the introduction part based on modules in logical algebras such as modules over JU algebras and some other structures and consequently add in the reference part. 2- We Shift the definition of bounded modules from introduction part to Preliminaries part. Definitions 2.1, 2.27 3- We Revise the English once again throughout the paper. See the authors' detailed response to the review by Moin A Ansari See the authors' detailed response to the review by Annet Kyomuhangi READ REVIEWER RESPONSES 1. Introduction The notion of bounded module was studied by Carl Faith. 1 Moreover, bounded submodule introduced with details by AL-LNI. 2 The concept of almost bounded submodule was submitted by Buthyna Najad. 3 Recently, restricted bounded submodule was studied by Mohammed Murad. 7 Also, modules in logical algebras such as modules over JU algebras and some other structures were introduced. 18 The scalar modules and prime modules are involved in several properties in our study 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 investigated how S-PS.B. modules relate to other well-known module types offering new insights into these connections. We explored 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 Definition 2.1 1 An F − module M is bounded module if there exists an element x ∈ M such that ann F ( M ) = ann F ( x ) . Definition 2.2 6 An 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 . Definition 2.3 7 An 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, An 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 . Definition 2.4 8 An 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 ) . Definition 2.5 9 An 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, an F − module M is said to be quasi-Dedekind if for each non-zero endomorphism of M is a F − monomorphism . Definition 2.6 10 An F − module M is called polyform if every essential submodule ℵ of M is dense. Note that every monoform is polyform. Definition 2.7 10 An 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 ) . Definition 2.8 10 An 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 . Definition 2.9 11 An F − module M is called coprime if ann F ( M ) = ann F ( M / ℵ ) for every proper submodule ℵ of M . Corollary 2.10 6 If M is a multiplication finitely generated F − module , then M is a scalar F − module . Remark 2.11 6 Let M be an injective scalar F − module . Then ℵ is a scalar submodule of M . Remark 2.12 12 Every finitely generated F − module is finitely annihilated. Proposition 2.13 12 Let M be a multiplication F − module . Then M is finitely generated if and only if M is finitely annihilated. Proposition 2.14 6 Let M be a scalar torsion-free F − module with ( F is an integral domain). Then every φ ∈ End ( M ) is F − monomorphism . Proposition 2.15 13 Let M be a quasi-Dedekind retractable F − module . If every 0 ≠ φ ∈ End ( M ) is a monomorphism, then M is compressible module. Proposition 2.16 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. Proposition 2.17 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. Proposition 2.18 14 Let M be a fully retractable F − module with End( M ) is a domain. Then M is polyform. Definition 2.19 15 An F − module M is called Rickart F − module if and only if Kerφ is a direct summand of M . Proposition 2.20 15 If M is an injective prime F − module , then M is Rickart module. Definition 2.21 16 An F − module M is called N-Rickart if for every homomorphism g : M → ℵ , ker g is a summand of M . Corollary 2.22 14 If M is uniform module, then M is fully polyform if and only if M is monoform module. Corollary 2.23 17 If M is finitely generated F − module , then M is compressible F − module if and only if M is uniform prime module. Definition 2.24 14 A module is called compressible module if it can be embedded in any of its nonzero submodules. Definition 2.25 14 A compressible module is called critically compressible if it cannot be embedded in any proper factor module. Definition 2.26 14 A partial endomorphism of a module M is a homomorphism from a submodule of M into M . Definition 2.27 3 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. 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 . Definition 3.1 A proper submodule ℵ of an 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 ) ) . Examples 3.2 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 established 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. Definition 3.3 An F − module M is said to be S-PS.B. F − module if every proper submodule of M is S-PS.B F − submodule. Examples 3.4 1- 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 . 2- 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 . Remark 3.5 More properties of S.PS.B. module (submodule): (i) Let M 1 and M 2 be two S-PS.B. F-modules, then M 1 ⨁ M 2 is S-PS.B. F-module. (ii) Let M be an F-module and I⊆ 〖 ann 〗 F ( M ) where I is an ideal of F. Then M is S-PS.B. F-module if and only if M is S-PS.B. F/I-module. (iii) If ℵ 1 and ℵ 2 are S-PS.B. F-submodules of M 1 and M 2 respectively, then ℵ 1 ⨁ℵ 2 is S-PS.B. F-submodule of M=M 1 ⨁M 2 . (iv) Let M be an F-module. If M is S-PS.B.E-module, then M is S-PS.B.F-module, where E=End(M). 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 got more results through several relationships. Proposition 4.1 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. Proof: ( 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. Corollary 4.2 Let M be a torsion-free S-PS.B. F − module . Then M is monoform module. Proof: 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 defined 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. Proposition 4.3 If M is quasi-Dedekind S-PS.B. F − module , then M is monoform module. Proof: 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 , 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. Proposition 4.4 Every S-PS.B. F − module is retractable F − module . Proof: 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 defined as φ ( x ) = tx , 0 ≠ x ∈ M , φ ( x ) ∈ ℵ with 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 . Proposition 4.5 If M is S-PS.B. torsion-free module, then M is critically compressible module. Proof: By Corollary (4.2) and Proposition (2.17) we get M is critically compressible. Corollary 4.6 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. Proposition 4.7 If M is a duo S-PS.B. F − module , then M is fully retractable module. Proof: 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. Corollary 4.8 If M is a duo S-PS.B. F − module and End ( M ) is a domain, then M is polyform module. Proof: 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. Proposition 4.9 Let M be a 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. Proof: ( 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 by 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. Corollary 4.10 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. Proof: If M is uniform, then polyform and fully polyform are equivalent, by Ref. 10 . Proposition 4.11 If M is a compressible F − module , then M is S-PS.B. F − module . Proof: 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 . Proposition 4.12 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. Proof: 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. Corollary 4.13 If M is a quasi-Dedekind retractable F − module , then M is S-PS.B. F − module . Proof: Since M is a quasi-Dedekind, then every F − homomorphism φ ∈ End ( M ) is monomorphism. Hence M is a compressible module, by Proposition (2.15) and thus M is S-PS.B. F − module , by Proposition (4.11) . Corollary 4.14 If M is a finitely generated, then every uniform prime module is S-PS.B. F − module . Proof: Since M is a finitely generated implies that M is a compressible module, by (2.23). The result is obtained by Proposition (4.11) . Proposition 4.15 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 . Proof: 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). Proposition 4.16 Let M is a quasi-Dedekind S-PS.B. F − module . Then ann F ( M ) ≠ ann F ( M ℵ ) , ℵ ≤ M . Proof: 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 ) . Corollary 4.17 Let M is a quasi-Dedekind S-PS.B. F − module . Then M is not coprime F − module . Proposition 4.18 Let ℵ ≤ M and M / ℵ be a quasi-Dedekind F − module . Then M is S-PS.B. F − module . Proof: Suppose that φ ∈ End ( M ) and φ : M ⟶ M defined as φ ( n ) = tn , n ∈ M , t ∈ F . 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 , t ∈ F 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 ) . Proposition 4.19 If M is S-PS.B. F − module , then M ≠ t M . Proof: Assume that M = t M and ∃ φ ∈ End ( M ) such that φ : M ⟶ M defined as φ ( x ) = tx , ∀ x ∈ M , t ∈ F , 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 . Proposition 4.20 Let M is S-PS.B. F − module such that 0 ≠ φ ∈ End ( M ) . Then φ is not epimorphism. Proof: 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. Proposition 4.21 Let M be S-PS.B. torsion-free F − module . Then M is an injective. Proof: Since M is 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 . Proposition 4.22 If M is a torsion-free and S-PS.B. F − module , then End ( M ) has no zero-divisors. Proof: 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. Proposition 4.23 If M is torsion-free S-PS.B., then M is a Rickart F − module . Proof: 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. Proposition 4.24 Let M be an indecomposable S-PS.B. F − module with M is N-Rickart. Then M is quasi-Dedekind. Proof: 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. Proposition 4.25 Let M be a torsion-free S-PS.B. Then for each φ ∈ End ( M ) there exists once φ is splits in M . Proof: 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 . Proposition 4.26 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. Proof: 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 we explained 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. Data availability 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 1. Faith C: Algebra II Ring Theory: Vol. 2: Ring Theory. Vol. 191.Springer Science & Business Media; 2012. Publisher Full Text 2. Mahmood LS, Al-Ani AS: Bounded Modules. Ibn Al-Haitham J. Pure Appl. Sci. 2017; 19 (3): 75–91. Reference Source 3. Shihab BN: On Almost Bounded Submodules. IIbn Al-Haitham J. Pure Appl. Sci. 2017; 23 (2): 168–174. Reference Source 4. Ramadhan HA, Al Mothafar NS: Weakly Small Semiprime Submodules. J. Phys. Conf. Ser. 2021; 1879 (3): 032128. IOP Publishing. Publisher Full Text 5. Hadi IMA, Al-Aeashi SN, Shyaa FD: T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules. Baghdad Sci. J. 2021; 18 (1): 0156. Publisher Full Text 6. Shihab BN: Scalar Reflexive Modules. 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Publisher Full Text Comments on this article Comments (0) Version 3 VERSION 3 PUBLISHED 09 Dec 2025 ADD YOUR COMMENT Comment Author details Author details 1 Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 31001, Iraq 2 Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, 10001, Iraq Amal Madhi Rashid Roles: Writing – Review & Editing Buthyna Najad Shihab Roles: Writing – Original Draft Preparation Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (3) version 3 Revised Published: 16 May 2026, 14:1385 https://doi.org/10.12688/f1000research.172196.3 version 2 Revised Published: 17 Apr 2026, 14:1385 https://doi.org/10.12688/f1000research.172196.2 version 1 Published: 09 Dec 2025, 14:1385 https://doi.org/10.12688/f1000research.172196.1 Copyright © 2026 Madhi Rashid A and Najad Shihab B. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Madhi Rashid A and Najad Shihab B. New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.12688/f1000research.172196.3 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 3 VERSION 3 PUBLISHED 16 May 2026 Revised Views 0 Cite How to cite this report: Ansari MA. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.201107.r485381 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v3#referee-response-485381 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 18 May 2026 Moin A Ansari , Jazan University, Jazan, Jazan, Saudi Arabia Approved VIEWS 0 https://doi.org/10.5256/f1000research.201107.r485381 I have gone through the revised version and found that it ... Continue reading READ ALL I have gone through the revised version and found that it is well revised now. I suggest to indexed this paper. Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebras and Its Applications. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Ansari MA. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.201107.r485381 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v3#referee-response-485381 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Version 2 VERSION 2 PUBLISHED 17 Apr 2026 Revised Views 0 Cite How to cite this report: K P S. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476796 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476796 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 11 May 2026 Shanmugapriya K P , Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, Tamil Nadu, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.198411.r476796 The article introduces a new class of modules called SSS-pseudo bounded modules over commutative rings with identity. The study investigates the relationships between this new module class and several well-known modules, including monoform, quasi-Dedekind, compressible, retractable, and ... Continue reading READ ALL The article introduces a new class of modules called SSS-pseudo bounded modules over commutative rings with identity. The study investigates the relationships between this new module class and several well-known modules, including monoform, quasi-Dedekind, compressible, retractable, and injective modules. Using module homomorphisms and the endomorphism ring End(M)End(M)End(M), the authors establish various properties, conditions, and corollaries related to SSS-pseudo bounded modules. The paper contributes to the development of module theory by extending existing algebraic concepts and providing new connections among generalized module structures. The work is mathematically relevant and may encourage further research in abstract algebra and module theory. Cite recent papers related to generalized module classes. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy set, Fuzzy algebra, Interval-Valued set, Intuitionistic Fuzzy set, Neutrosophic set, Cubic set I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT K P S. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476796 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476796 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Kalavath AN. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476804 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476804 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 07 May 2026 Anjaneyulu Naik Kalavath , Acharya Nagarjuna University, Guntur, Andhra Pradesh, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.198411.r476804 Thank you for the opportunity to peer review this manuscript. Manuscript Title: New Class of S-Pseudo Bounded Modules with Some Related Concepts The authors define a new class of modules, termed S-Pseudo Bounded ... Continue reading READ ALL Thank you for the opportunity to peer review this manuscript. Manuscript Title: New Class of S-Pseudo Bounded Modules with Some Related Concepts The authors define a new class of modules, termed S-Pseudo Bounded (S-PS.B.) modules. The definition appears to be original, and the authors provide clear examples and counterexamples that distinguish S-PS.B. modules from related notions such as Endo-R.B. modules. The paper is well situated within the existing literature, engaging with concepts such as bounded modules, monoform modules, compressible modules, retractable modules, and Rickart modules. The manuscript develops more than twenty propositions and corollaries that establish relationships between S-PS.B. modules and several well-known classes of modules, including monoform, compressible, retractable, and quasi-Dedekind modules. Overall, the arguments are rigorous, and the results are of interest to researchers in the area. However, the paper would benefit from a broader discussion of the structural properties of S-PS.B. modules. In particular, closure properties under quotients, products, direct summands, and extensions are not addressed. At present, only finite direct sums are considered (Remark 3.5). Expanding this discussion would strengthen the contribution. Strengths of this paper Introduces a new and well-defined class of modules, supported by illustrative examples and counterexamples. Establishes meaningful and non-trivial connections between S-PS.B. modules and several existing module classes. Some Revisions Required The Introduction would benefit from 2-3 sentences clarifying the motivation for S-PS.B. modules and distinguishing them more explicitly from Endo-R.B. modules. Add a title to the diagram on page 8 and reference it explicitly in the text, in line with F1000Research guidelines. Clarify the relationship between S-PS.B. modules and bounded or almost bounded modules. A careful proofreading is recommended to correct minor typographical and grammatical issues. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: Pure mathematics(Algebras), Fuzzy algebraic structures, hyper structures, soft structures, intuitionistic fuzzy sets and neutrosophic fuzzy sets with decision-making I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Kalavath AN. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476804 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476804 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Oduor MO. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476800 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476800 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 07 May 2026 Maurice Owino Oduor , Department of Mathematics, University of Kabianga, Kericho, Kericho County, Kenya Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.198411.r476800 General Comment: There is need to improve on the grammar. The authors should use language editor to improve on the construction of sentences, use of correct words/phrases and punctuation marks. Abstract The second sentence should read ... Continue reading READ ALL General Comment: There is need to improve on the grammar. The authors should use language editor to improve on the construction of sentences, use of correct words/phrases and punctuation marks. Abstract The second sentence should read as follows: We define a new class of F- module, namely, S-Pseudo bounded module denoted by S-PS. B. F - module and introduce different approaches to relate this class with other types of well- known modules such as ... 1. Introduction Paragraph 2. Line 3. Delete 'and connected' to read: we discovered some significant relationships with other modules. Definition 2.3 is not clear. Definition 2.5 is not clear. Corollary is a consequence of a theorem. So, Corollary 2.10 is a consequence of which theorem? Remark 2.11 is not clear. Restructure Proposition 2.18, by replacing the word 'with' by 'where'. In definition 2.19, what is varphi? Definition 2.21 should precede Definition 2.19 and Proposition 2.20. Corollary 2.22 is a consequence of which theorem? Corollary 2.23 is a consequence of which theorem? 3. S- Pseudo bounded modules I suggest that the number of examples under 3.2 to be reduced to two. Remark 3.5 should be renamed Theorem 3.5 and a precise proof be given. 4. S- Pseudo bounded modules with some modules The title sounds clumsy. This section should be merged with section 3. Proposition 4.1. In the proof where (ii) implies (i) to read: Given that M is a monoform module, then it is a uniform prime module... The last statement to read: However, the condition of torsion- free is sufficient to prove that M is monoform module. Proposition 4.9. Replace the word 'an' with 'a'. Proposition 4.21. In the proof, correct the spelling of the word 'monomorphism.' Further, a proof should not begin with the word 'Since' followed by the word 'thus'. Proposition 4.22. Replace the word 'implies' with the phrase 'so that'. Proposition 4.23. In the proof, delete the word 'an.' Further, a proof should not begin with the word 'Since' followed by the word 'thus'. Proposition 4.25 is not clear. The grammar in the proof is poor. 5. Conclusion. Poor grammar. To be revisited. There is need to include a section on Recommendations for further research. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? No Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: Commutative Algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Oduor MO. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476800 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476800 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Ansari MA. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476801 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476801 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 06 May 2026 Moin A Ansari , Jazan University, Jazan, Jazan, Saudi Arabia Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.198411.r476801 Some more recent works should be mentioned in the introduction part based on modules in logical algebras such as modules over JU algebras and some other structures. Mention them in introduction part and consequently add in the reference part. ... Continue reading READ ALL Some more recent works should be mentioned in the introduction part based on modules in logical algebras such as modules over JU algebras and some other structures. Mention them in introduction part and consequently add in the reference part. Shift the definition of bounded modules from introduction part to Preliminaries part. Revise the English once again throughout the paper. After these three modifications, I recommend accepting this article for publication in F1000Research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Ansari M, Iampan A, Haider A: Modules over JU-Algebras. Asia Pacific Journal of Mathematics . 2026; 13 . Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebras and Its Applications. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Ansari MA. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476801 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476801 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 08 May 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 08 May 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: no competing interests Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 08 May 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 08 May 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: no competing interests Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Yedlapalli P and Noorbhasha R. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476803 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476803 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 27 Apr 2026 Phani Yedlapalli , Shri Vishnu Engineering College for Women, Kovvada, Andhra Pradesh, India Rafi Noorbhasha , Department of Mathematics, Siddhartha Academy of Higher Education, Vijayawada, Andhra Pradesh, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.198411.r476803 The manuscript entitled “New Class of S-Pseudo Bounded Modules With Some Related Concepts” presents an original and valuable contribution to the theory of modules by introducing the notion of S-Pseudo Bounded modules and establishing meaningful relationships with several well-known module ... Continue reading READ ALL The manuscript entitled “New Class of S-Pseudo Bounded Modules With Some Related Concepts” presents an original and valuable contribution to the theory of modules by introducing the notion of S-Pseudo Bounded modules and establishing meaningful relationships with several well-known module classes. The paper is mathematically interesting, logically developed, and supported with appropriate examples, propositions, and structural results. The authors have satisfactorily revised the manuscript by incorporating additional properties of S-Pseudo Bounded modules, clarifying definitions, correcting notation, and addressing previous reviewer comments. These revisions have significantly improved the presentation and strengthened the overall quality of the work. The results are relevant to current research in module theory and may motivate further investigations in related algebraic structures. The manuscript is suitable for indexing after careful proofreading to correct minor typographical and grammatical errors that remain in the text. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebra We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Yedlapalli P and Noorbhasha R. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476803 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476803 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Version 1 VERSION 1 PUBLISHED 09 Dec 2025 Views 0 Cite How to cite this report: Kyomuhangi A. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.189909.r464812 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v1#referee-response-464812 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 02 Apr 2026 Annet Kyomuhangi , Busitema University, Tororo, Uganda Not Approved VIEWS 0 https://doi.org/10.5256/f1000research.189909.r464812 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 ... Continue reading READ ALL 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Kyomuhangi A. Reviewer Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.189909.r464812 ) The direct URL for this report is: https://f1000research.com/articles/14-1385/v1#referee-response-464812 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 17 Apr 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 17 Apr 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 17 Apr 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 17 Apr 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests Close Report a concern COMMENT ON THIS REPORT Comments on this article Comments (0) Version 3 VERSION 3 PUBLISHED 09 Dec 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 4 5 6 Version 3 (revision) 16 May 26 read Version 2 (revision) 17 Apr 26 read read read read read Version 1 09 Dec 25 read Annet Kyomuhangi , Busitema University, Tororo, Uganda Phani Yedlapalli , Shri Vishnu Engineering College for Women, Kovvada, India Rafi Noorbhasha , Siddhartha Academy of Higher Education, Vijayawada, India Moin A Ansari , Jazan University, Jazan, Saudi Arabia Maurice Owino Oduor , University of Kabianga, Kericho, Kenya Anjaneyulu Naik Kalavath , Acharya Nagarjuna University, Guntur, India Shanmugapriya K P , Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Ansari M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 18 May 2026 | for Version 3 Moin A Ansari , Jazan University, Jazan, Jazan, Saudi Arabia 0 Views copyright © 2026 Ansari M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions I have gone through the revised version and found that it is well revised now. I suggest to indexed this paper. Competing Interests No competing interests were disclosed. Reviewer Expertise Algebras and Its Applications. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Ansari MA. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.201107.r485381) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v3#referee-response-485381 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 K P S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 11 May 2026 | for Version 2 Shanmugapriya K P , Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, Tamil Nadu, India 0 Views copyright © 2026 K P S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The article introduces a new class of modules called SSS-pseudo bounded modules over commutative rings with identity. The study investigates the relationships between this new module class and several well-known modules, including monoform, quasi-Dedekind, compressible, retractable, and injective modules. Using module homomorphisms and the endomorphism ring End(M)End(M)End(M), the authors establish various properties, conditions, and corollaries related to SSS-pseudo bounded modules. The paper contributes to the development of module theory by extending existing algebraic concepts and providing new connections among generalized module structures. The work is mathematically relevant and may encourage further research in abstract algebra and module theory. Cite recent papers related to generalized module classes. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy set, Fuzzy algebra, Interval-Valued set, Intuitionistic Fuzzy set, Neutrosophic set, Cubic set I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) K P S. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476796) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476796 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Kalavath A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 07 May 2026 | for Version 2 Anjaneyulu Naik Kalavath , Acharya Nagarjuna University, Guntur, Andhra Pradesh, India 0 Views copyright © 2026 Kalavath A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Thank you for the opportunity to peer review this manuscript. Manuscript Title: New Class of S-Pseudo Bounded Modules with Some Related Concepts The authors define a new class of modules, termed S-Pseudo Bounded (S-PS.B.) modules. The definition appears to be original, and the authors provide clear examples and counterexamples that distinguish S-PS.B. modules from related notions such as Endo-R.B. modules. The paper is well situated within the existing literature, engaging with concepts such as bounded modules, monoform modules, compressible modules, retractable modules, and Rickart modules. The manuscript develops more than twenty propositions and corollaries that establish relationships between S-PS.B. modules and several well-known classes of modules, including monoform, compressible, retractable, and quasi-Dedekind modules. Overall, the arguments are rigorous, and the results are of interest to researchers in the area. However, the paper would benefit from a broader discussion of the structural properties of S-PS.B. modules. In particular, closure properties under quotients, products, direct summands, and extensions are not addressed. At present, only finite direct sums are considered (Remark 3.5). Expanding this discussion would strengthen the contribution. Strengths of this paper Introduces a new and well-defined class of modules, supported by illustrative examples and counterexamples. Establishes meaningful and non-trivial connections between S-PS.B. modules and several existing module classes. Some Revisions Required The Introduction would benefit from 2-3 sentences clarifying the motivation for S-PS.B. modules and distinguishing them more explicitly from Endo-R.B. modules. Add a title to the diagram on page 8 and reference it explicitly in the text, in line with F1000Research guidelines. Clarify the relationship between S-PS.B. modules and bounded or almost bounded modules. A careful proofreading is recommended to correct minor typographical and grammatical issues. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise Pure mathematics(Algebras), Fuzzy algebraic structures, hyper structures, soft structures, intuitionistic fuzzy sets and neutrosophic fuzzy sets with decision-making I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Kalavath AN. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476804) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476804 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Oduor M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 07 May 2026 | for Version 2 Maurice Owino Oduor , Department of Mathematics, University of Kabianga, Kericho, Kericho County, Kenya 0 Views copyright © 2026 Oduor M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions General Comment: There is need to improve on the grammar. The authors should use language editor to improve on the construction of sentences, use of correct words/phrases and punctuation marks. Abstract The second sentence should read as follows: We define a new class of F- module, namely, S-Pseudo bounded module denoted by S-PS. B. F - module and introduce different approaches to relate this class with other types of well- known modules such as ... 1. Introduction Paragraph 2. Line 3. Delete 'and connected' to read: we discovered some significant relationships with other modules. Definition 2.3 is not clear. Definition 2.5 is not clear. Corollary is a consequence of a theorem. So, Corollary 2.10 is a consequence of which theorem? Remark 2.11 is not clear. Restructure Proposition 2.18, by replacing the word 'with' by 'where'. In definition 2.19, what is varphi? Definition 2.21 should precede Definition 2.19 and Proposition 2.20. Corollary 2.22 is a consequence of which theorem? Corollary 2.23 is a consequence of which theorem? 3. S- Pseudo bounded modules I suggest that the number of examples under 3.2 to be reduced to two. Remark 3.5 should be renamed Theorem 3.5 and a precise proof be given. 4. S- Pseudo bounded modules with some modules The title sounds clumsy. This section should be merged with section 3. Proposition 4.1. In the proof where (ii) implies (i) to read: Given that M is a monoform module, then it is a uniform prime module... The last statement to read: However, the condition of torsion- free is sufficient to prove that M is monoform module. Proposition 4.9. Replace the word 'an' with 'a'. Proposition 4.21. In the proof, correct the spelling of the word 'monomorphism.' Further, a proof should not begin with the word 'Since' followed by the word 'thus'. Proposition 4.22. Replace the word 'implies' with the phrase 'so that'. Proposition 4.23. In the proof, delete the word 'an.' Further, a proof should not begin with the word 'Since' followed by the word 'thus'. Proposition 4.25 is not clear. The grammar in the proof is poor. 5. Conclusion. Poor grammar. To be revisited. There is need to include a section on Recommendations for further research. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? No Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise Commutative Algebra I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Oduor MO. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476800) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476800 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Ansari M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 06 May 2026 | for Version 2 Moin A Ansari , Jazan University, Jazan, Jazan, Saudi Arabia 0 Views copyright © 2026 Ansari M. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Some more recent works should be mentioned in the introduction part based on modules in logical algebras such as modules over JU algebras and some other structures. Mention them in introduction part and consequently add in the reference part. Shift the definition of bounded modules from introduction part to Preliminaries part. Revise the English once again throughout the paper. After these three modifications, I recommend accepting this article for publication in F1000Research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Ansari M, Iampan A, Haider A: Modules over JU-Algebras. Asia Pacific Journal of Mathematics . 2026; 13 . Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Algebras and Its Applications. I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (1) Author Response 08 May 2026 Amal Rashid, Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 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. View more View less Competing Interests no competing interests reply Respond Report a concern Ansari MA. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476801) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476801 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Yedlapalli P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 27 Apr 2026 | for Version 2 Phani Yedlapalli , Shri Vishnu Engineering College for Women, Kovvada, Andhra Pradesh, India Rafi Noorbhasha , Department of Mathematics, Siddhartha Academy of Higher Education, Vijayawada, Andhra Pradesh, India 0 Views copyright © 2026 Yedlapalli P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The manuscript entitled “New Class of S-Pseudo Bounded Modules With Some Related Concepts” presents an original and valuable contribution to the theory of modules by introducing the notion of S-Pseudo Bounded modules and establishing meaningful relationships with several well-known module classes. The paper is mathematically interesting, logically developed, and supported with appropriate examples, propositions, and structural results. The authors have satisfactorily revised the manuscript by incorporating additional properties of S-Pseudo Bounded modules, clarifying definitions, correcting notation, and addressing previous reviewer comments. These revisions have significantly improved the presentation and strengthened the overall quality of the work. The results are relevant to current research in module theory and may motivate further investigations in related algebraic structures. The manuscript is suitable for indexing after careful proofreading to correct minor typographical and grammatical errors that remain in the text. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Algebra We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Yedlapalli P and Noorbhasha R. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.198411.r476803) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v2#referee-response-476803 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Kyomuhangi A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 02 Apr 2026 | for Version 1 Annet Kyomuhangi , Busitema University, Tororo, Uganda 0 Views copyright © 2026 Kyomuhangi A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Not Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes 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 Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (1) Author Response 17 Apr 2026 Amal Rashid, Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 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. View more View less Competing Interests No competing interests reply Respond Report a concern Kyomuhangi A. Peer Review Report For: New Class of S-Pseudo Bounded Modules With Some Related Concepts [version 3; peer review: 3 approved, 2 approved with reservations, 1 not approved] . F1000Research 2026, 14 :1385 ( https://doi.org/10.5256/f1000research.189909.r464812) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1385/v1#referee-response-464812 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. 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