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Some properties of S-Pseudo bounded radical are studied. A proper submodule ℵ of an F − module M is said to be S-pseudo bounded submodule if there exists , φ ∈ End ( M ) , x ∈ M such that φ ( x ) ∈ ℵ implies ann F ( ℵ ) = ann F φ ( x ) making use of endomorphism map over an F − module . The characterization of S-Pseudo bounded radical for finitely generated and multiplication module is given. It must be emphasized that scalar module and prime submodule played a major role in achieving new results and to study the relationship between the radical of submodules and the radical of S-pseudo bounded submodule. In our work many properties and corollaries will be proved which explain the idea of the radical of S-pseudo bounded submodule and a new class of F − module as well as F − submodule provided with some examples that illustrate and clarify in a nice way this type of module (submodule). We used the symbol End ( M ) which means the set of all endomorphism maps of F − module M and S − rad M PS . B . ( ℵ ) refers to the intersection of all S-pseudo bounded submodules of M containing ℵ ." } { "@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/14-1378/v2", "name": "S-Pseudo Bounded Radical of Submodules" } } ] } Home Browse S-Pseudo Bounded Radical of Submodules ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Madhi Rashid A and Najad Shihab B. S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.12688/f1000research.172188.2 ) 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 S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 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 20 Feb 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 this paper, every module M is unitary and every ring F is commutative with identity. Some properties of S-Pseudo bounded radical are studied. A proper submodule ℵ of an F − module M is said to be S-pseudo bounded submodule if there exists , φ ∈ End ( M ) , x ∈ M such that φ ( x ) ∈ ℵ implies ann F ( ℵ ) = ann F φ ( x ) making use of endomorphism map over an F − module . The characterization of S-Pseudo bounded radical for finitely generated and multiplication module is given. It must be emphasized that scalar module and prime submodule played a major role in achieving new results and to study the relationship between the radical of submodules and the radical of S-pseudo bounded submodule. In our work many properties and corollaries will be proved which explain the idea of the radical of S-pseudo bounded submodule and a new class of F − module as well as F − submodule provided with some examples that illustrate and clarify in a nice way this type of module (submodule). We used the symbol End ( M ) which means the set of all endomorphism maps of F − module M and S − rad M PS . B . ( ℵ ) refers to the intersection of all S-pseudo bounded submodules of M containing ℵ . READ ALL READ LESS Keywords S-pseudo bounded radical, Finitely generated module, Prime submodule, Multiplication modules, 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. S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.12688/f1000research.172188.2 ) First published: 09 Dec 2025, 14 :1378 ( https://doi.org/10.12688/f1000research.172188.1 ) Latest published: 17 Apr 2026, 14 :1378 ( https://doi.org/10.12688/f1000research.172188.3 ) Revised Amendments from Version 1 1. The Abstract has been amended as required. 2.The definition has been corrected by replacing the slash with a vertical line. 3.The numbering of examples has been separated from that of Definitions and Theorems, And make the necessary adjustments 4.In Lemma 2.4 (Proof), the phrase “In the same way” has been replaced with “Similarly”. 5. The beginning of section 3 has been modified. 6. The proof was modified in 3.5, 3.7, 3.11, 3.12, 3,19. 7.In Proposition 3.10, the spelling of “epimorphism” has been corrected. 8.The notation defined below the proof of Proposition 3.20 has been moved to Section 1 (Introduction), and the word “denoted” has been replaced with “denotes”. 9.Corollary 3.23 has been revised to read: “… and … are ideals of …. Then…” 10.In the Conclusion section, the word “several” has been deleted, and the opening sentence has been restructured as follows) In this study, we have re-discovered and proved some properties of S-pseudo bounded radical submodules”. 11.The definition 2.1 has been refined to ensure greater clarity and precision also the definition in Abstract. 12.The proof of Lemm2.4 has been reformulated, and all spelling errors have been corrected. 13.Each symbol has been explicitly defined, with its meaning clearly stated. 14.the final statement (where (ℵ:M) be a prime ideal of F ) of proposition 3.8 is an assumption not a derived conclusion. 1. The Abstract has been amended as required. 2.The definition has been corrected by replacing the slash with a vertical line. 3.The numbering of examples has been separated from that of Definitions and Theorems, And make the necessary adjustments 4.In Lemma 2.4 (Proof), the phrase “In the same way” has been replaced with “Similarly”. 5. The beginning of section 3 has been modified. 6. The proof was modified in 3.5, 3.7, 3.11, 3.12, 3,19. 7.In Proposition 3.10, the spelling of “epimorphism” has been corrected. 8.The notation defined below the proof of Proposition 3.20 has been moved to Section 1 (Introduction), and the word “denoted” has been replaced with “denotes”. 9.Corollary 3.23 has been revised to read: “… and … are ideals of …. Then…” 10.In the Conclusion section, the word “several” has been deleted, and the opening sentence has been restructured as follows) In this study, we have re-discovered and proved some properties of S-pseudo bounded radical submodules”. 11.The definition 2.1 has been refined to ensure greater clarity and precision also the definition in Abstract. 12.The proof of Lemm2.4 has been reformulated, and all spelling errors have been corrected. 13.Each symbol has been explicitly defined, with its meaning clearly stated. 14.the final statement (where (ℵ:M) be a prime ideal of F ) of proposition 3.8 is an assumption not a derived conclusion. See the authors' detailed response to the review by Maurice Owino Oduor See the authors' detailed response to the review by Alaa Abouhalaka and Hassan Alhussein READ REVIEWER RESPONSES  There is a newer version of this article available. Suppress this message for one day. 1. Introduction The concept of the radical of an ideal plays important role in the study of rings and it was generalized to modules over commutative rings. 1 , 2 A submodule ℵ of M is called prime submodule if ℵ ≠ M and given t ∈ F , m ∈ M , tm ∈ ℵ implies m ∈ ℵ or t ∈ ( ℵ : M ) where ( ℵ : M ) = { t ∈ F | t M ⊆ ℵ } , if ℵ is prime then ( ℵ : M ) is prime ideal of F . 3 – 5 The M − radical of a submodule ℵ was defined as the intersection of all prime submodules of M containing ℵ , if there is no prime submodule containing ℵ then rad ( ℵ ) = M . 6 , 7 For a proper submodule ℵ of an F − module M , the intersection of all S-PS.B. submodules of M containing ℵ is called the radical of ℵ and denotes by S − rad M PS . B . ( ℵ ) , if there is no S-PS.B. submodule of M containing ℵ , then S − rad M PS . B . ( ℵ ) = M . A submodule ℵ of M is called a radical submodule if S − rad M PS . B . ( ℵ ) = ℵ . The radical of an ideal I of F has the characterization I = { x ∈ F | x m ∈ I for some m > 0 } . 8 It was mentioned to completely irreducible submodule where a submodule ℵ of an F − module M is named completely irreducible if for any two submodules ℵ 1 ,ℵ 2 of M such that ℵ 1 ∩ ℵ 2 ⊆ ℵ implies that ℵ 1 ⊆ ℵ or ℵ 2 ⊆ ℵ. 9 , 10 The primary objective of this study is to present novel and diverse results concerning the S-pseudo bounded radical of submodules, and to establish a characterization for this concept, which is regarded as a new notion. We explore the relationship between the radical of submodules and that of S-pseudo bounded submodules. This connection yields significant results regarding their radicals. In the following section, we provide definitions, examples, and lemmas to further clarify our study. 2. S-Pseudo bounded submodules At the beginning of this section, it is essential to introduce the definition of an S-pseudo bounded submodule along with illustrative examples. Additionally, it is necessary to present several lemmas that play a significant role in the study of the S-pseudo bounded radical of submodules. Definition 2.1 A proper submodule ℵ of an F − module M is said to be S-pseudo bounded submodule if there exists φ ∈ S = End ( M ) such that φ ( x ) ∈ ℵ , for all x ∈ M implies that ann F ( ℵ ) = ann F φ ( x ) , for some x ∈ M , where End ( M ) is the set of endomorphism maps over an F − module M . Examples 1 1- Let M = ℤ 2 ⨁ ℤ 4 as a ℤ − module . Define φ : M ⟶ M be φ ( a ¯ , b ¯ ) = ( 0 , ¯ b ¯ ) , ∀ ( a ¯ , b ¯ ) ∈ M , if ℵ = ⟨ 0 ¯ ⟩ ⨁ ℤ 4 , then φ ( a ¯ , b ¯ ) ∈ ℵ . Hence ℵ is S-PS.B. ℤ − submodule of M , since ( a ¯ , b ¯ ) = ( 0 ¯ , 2 ¯ ) ∈ M implies that ann ℤ ( ℵ ) = ann ℤ ( 0 ¯ , 2 ¯ ) = 2 ℤ . 2- Consider M = ℤ ⨁ ℤ 3 as a ℤ − module . Define φ : M ⟶ M be φ ( a , b ¯ ) = ( 0 , 0 ¯ ) , ∀ ( a , b ¯ ) ∈ M , if ℵ = 3 ℤ ⨁ ⟨ 0 ¯ ⟩ , it is clear φ ∈ End ( M ) . Let ( a , b ¯ ) = ( 3 , 0 ¯ ) ∈ M , hence ann ℤ ( ℵ ) = ann ℤ ( 3 ℤ ⨁ ⟨ 0 ¯ ⟩ ) = ⟨ 0 ¯ ⟩ , but ann ℤ φ ( 3 , 0 ¯ ) = ann ℤ ( 0 , 0 ¯ ) = ℤ . Thus ℵ is not S-PS.B. ℤ − submodule of M . Definition 2.2 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. Lemma 2.3 If ℵ is a prime submodule of a scalar F − module M , then ℵ is S-PS.B. F − submodule . Proof: Let φ ∈ End ( M ) , x ∈ M . Since M is scalar F − module , then for all φ ∈ End ( M ) , ∃ 0 ≠ r ∈ F such that φ ( x ) = rx , for all x ∈ M . Now, we must show that ann F φ ( x ) = ann F ℵ . Let a ∈ ann F φ ( x ) , x ∈ M . Then a n ∈ ann F φ ( x ) , for some n ∈ ℤ + and a n . φ ( x ) = 0 , so a n . rx = 0 implies that a n ∈ ann F ( rx ) , for all rx ∈ ℵ . Since ℵ is a prime submodule, then either x ∈ ℵ or r M ⊆ ℵ so that a n ∈ ann F ℵ and a ∈ ann F ℵ . Similarly, ann F ℵ ⊆ ann F φ ( x ) . Hence, ℵ is S-PS.B. F − submodule . Lemma 2.4 If ℵ is S-PS.B. F − submodule of a scalar F − module M , then ℵ is S-prime submodule. Proof: Let φ ∈ End ( M ) and define φ : M ⟶ M be φ ( x ) = rx ,∀x ∈ M . Let x ∉ ℵ, then we have to prove φ ( M ) ⊆ ℵ. Since M is scalar and by definition of S-PS.B submodule, we have φ( x ) ∈ ℵ,∀ x ∈ M , which means that φ ( M ) ⊆ ℵ. Lemma 2.5 If ℵ is a submodule of a scalar F − module M , then S − rad M PS . B . ( ℵ ) = rad ( ℵ ) . Proof: Note that every S-prime submodule is prime, then from previous lemmas we get the result. 3. Some results related to S-pseudo bounded radical of submodules We have examined the S-pseudo bounded radical of submodules presenting various properties exploring several relationships and introducing new results. Definition 3.1 A S-pseudo bounded radical of a submodule ℵ of an F − module M is the intersection of all S-pseudo bounded submodules of M containing ℵ and denotes by S − rad M PS . B . ( ℵ ) . If there is no S-PS.B. submodule of M contains ℵ , then S − rad M PS . B . ( ℵ ) = M . Definition 3.2 A proper submodule ℵ of an F − module M is called S-pseudo bounded radical submodule if S − rad M PS . B . ( ℵ ) = ℵ . Note that S-pseudo bounded radical submodule is proper submodule of M . Remark 3.3 If ℵ is S-PS.B. submodule of an F − module M , then S − rad M PS . B . ( ℵ ) is S-PS.B. submodule too. Proof: Using the Definition (3.1) , we have S − rad M PS . B . ( ℵ ) = ∩ j ∈ J { k j : k j is S − PS . B . submodule , ℵ ⊆ k j } , for j ∈ J and J be an index set. Then by using induction and properties of S-PS.B. submodule we get S − rad M PS . B . ( ℵ ) is S-PS.B. submodule. Proposition 3.4 If ℵ , K are two submodules of an F − module M . Then (i) ℵ ⊆ S − rad M PS . B . ( ℵ ) . (ii) If ℵ ⊆ K , then S − rad M PS . B . ( ℵ ) ⊆ S − rad M PS . B . ( K ) . (iii) S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) ) = S − rad M PS . B . ( ℵ ) . Proof: (i) By the Definition (3.1) , we have S − rad M PS . B . ( ℵ ) = ∩ j ∈ J k j , j ∈ J and J be an index set and S-PS.B. submodules k j of M containing ℵ so that ℵ ⊆ S − rad M PS . B . ( ℵ ) . (ii) Let L j be an S-PS.B. submodule of an F − module M such that K ⊆ L j , j ∈ J and J be an index set. Then ℵ ⊆ K ⊆ L j so that ℵ ⊆ L j and by Definition (3.1) , we get S − rad M PS . B . ( ℵ ) ⊆ S − rad M PS . B . ( K ) . (iii) From Definition (3.1) , we have S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) ) = ∩ j ∈ J G j where G j is an S-PS.B. submodules of M such that S − rad M PS . B . ( ℵ ) ⊆ G j . By part ( i ) , we get ℵ ⊆ S − rad M PS . B . ( ℵ ) , thus S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) ) ⊆ S − rad M PS . B . ( ℵ ) . From part ( ii ) , we obtain S − rad M PS . B . ( ℵ ) ⊆ S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) ) . Proposition 3.5 If ℵ j is a submodule of an F − module M , J is an index set, j ∈ J , then (i) S − rad M PS . B . ( ∩ j ∈ J ℵ j ) ⊆ ∩ j ∈ J S − rad M PS . B . ( ℵ j ) = S − rad M PS . B . ( ∩ j ∈ J S − rad M PS . B . ( ℵ j ) ) . (ii) ∑ j ∈ J S − rad M PS . B . ( ℵ j ) ⊆ S − rad M PS . B . ( ∑ j ∈ J . ℵ j ) = S − rad M PS . B . ( ∑ j ∈ J . S − rad M PS . B . ( ℵ j ) ) . Proof: (i) Suppose S − rad M PS . B . ( ∩ j ∈ J ℵ j ) = ∩ k ∈ K V k where V k is S-PS.B. submodule containing ∩ j ∈ J ℵ j for k ∈ K , K be an index set. Let { G kj } be the set of S-PS.B. submodules containing ℵ j . Since ∩ j ∈ J ℵ j ⊆ ℵ j ⊆ G kj , then S − rad M PS . B . ( ∩ j ∈ J ℵ j ) ⊆ S − rad M PS . B . ( ℵ j ) . Since ∩ j ∈ J S − rad M PS . B . ( ℵ j ) ⊆ S − rad M PS . B . ( ℵ j ) , then S − rad M PS . B . ( ∩ j ∈ J S − rad M PS . B . ( ℵ j ) ) ⊆ S − rad M PS . B . ( S − rad M PS . B . ( ℵ j ) ) = S − rad M PS . B . ( ℵ j ) , j ∈ J . Hence S − rad M PS . B . ( ∩ j ∈ J S − rad M PS . B . ( ℵ j ) ) ⊆ ∩ j ∈ J S − rad M PS . B . ( ℵ j ) . By previous proposition part ( i ) , ∩ j ∈ J S − rad M PS . B . ( ℵ j ) = S − rad M PS . B . ( ∩ j ∈ J S − rad M PS . B . ( ℵ j ) ) is clear. (ii) Since ℵ j ⊆ ∑ j ∈ J ℵ j for j ∈ J , then from previous proposition part (ii) S − rad M PS . B . ( ℵ j ) ⊆ S − rad M PS . B . ( ∑ j ∈ J ℵ j ) . Hence ∑ j ∈ J S − rad M PS . B . ( ℵ j ) ⊆ S − rad M PS . B . ( ∑ j ∈ J ℵ j ) . Since ∑ j ∈ J ℵ j ⊆ ∑ j ∈ J S − rad M PS . B . ( ℵ j ) , then it is easy to show that S − rad M PS . B . ( ∑ j ∈ J ℵ j ) = S − rad M PS . B . ( ∑ j ∈ J S − rad M PS . B . ( ℵ j ) ) . Proposition 3.6 If ℵ is a submodule of a multiplication finitely generated F − module M . Then S − rad M PS . B . ( ℵ ) = M if and only if ℵ = M . Proof: Assume that S − rad M PS . B . ( ℵ ) = M and ℵ ≠ M . Since M is a finitely generated module, then there exists a maximal submodule K of M such that ℵ ⊆ K and hence K is prime. Since M is a finitely generated multiplication F − module, then M is scalar 11 and thus K is S-PS.B. submodule Lemma (2.34) . Thus S − rad M PS . B . ( ℵ ) ⊆ K so that M = K which is contradicting the assumption. If ℵ = M , the other side is clear. Corollary 3.7 Let ℵ , K be two submodules of a multiplication finitely generated F − module M . Then S − rad M PS . B . ( ℵ ) + S − rad M PS . B . ( K ) = M if and only if ℵ + K = M . Proof: Assume that S − rad M PS . B . ( ℵ ) + S − rad M PS . B . ( K ) = M , then by proposition (3.4) , (3.5) and previous proposition, M = S − rad M PS . B . ( ℵ + K ) implies that ℵ + K = M . On the other hand if ℵ + K = M , then M = S − rad M PS . B . ( ℵ + K ) = S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) + S − rad M PS . B . ( K ) ) . Hence by previous proposition, S − rad M PS . B . ( ℵ ) + S − rad M PS . B . ( K ) = M . Proposition 3.8 If M be a multiplication finitely generated F − module and ℵ be a submodule of M , then S − rad M PS . B . ( ℵ ) = S − rad M PS . B . ( ℵ + ( ℵ : M ) M ) , where ( ℵ : M ) be a prime ideal of F . Proof: Since ℵ ⊆ ℵ + ( ℵ : M ) M , then S − rad M PS . B . ( ℵ ) ⊆ S − rad M PS . B . ( ℵ + ( ℵ : M ) M ) . Assume that S − rad M PS . B . ( ℵ ) = ∩ j ∈ J V j where V j is S-PS.B. submodule of M containing ℵ . Since M is a finitely generated multiplication module, then M is scalar 11 and thus ℵ is prime Lemma (2.4) so that V j is prime submodule and ( V j : M ) is prime ideal which implies that ( ℵ : M ) ⊆ ( V j : M ) and hence ℵ + ( ℵ : M ) M ⊆ V j . Therefore S − rad M PS . B . ( ℵ + ( ℵ : M ) M ) ⊆ ∩ j ∈ J V j = S − rad M PS . B . ( ℵ ) . Hence S − rad M PS . B . ( ℵ ) = S − rad M PS . B . ( ℵ + ( ℵ : M ) M ) . Corollary 3.9 If M is a multiplication finitely generated F − module and ℵ is a submodule of M , then S − rad M PS . B . ( ℵ ) = ℵ + ( ℵ : M ) M , where ( ℵ : M ) is a maximal ideal of F . Proof: Since ℵ + ( ℵ : M ) M ⊆ S − rad M PS . B . ( ℵ + ( ℵ : M ) M ) = S − rad M PS . B . ( ℵ ) , by previous proposition and since ( ℵ : M ) ⊆ ( ( ℵ + ( ℵ : M ) M ) : M ) , then ( ( ℵ + ( ℵ : M ) M ) : M ) = ( ℵ : M ) or ( ( ℵ + ( ℵ : M ) M ) : M ) = F . If ( ( ℵ + ( ℵ : M ) M ) : M ) = F , then FM = M ⊆ ℵ + ( ℵ : M ) M ⊆ S − rad M PS . B . ( ℵ ) which implies that M = S − rad M PS . B . ( ℵ ) . Since M is finitely generated, then M = ℵ by Proposition (3.6) which is contradiction. So ( ( ℵ + ( ℵ : M ) M ) : M ) = ( ℵ : M ) , therefore S − rad M PS . B . ( ℵ ) = ℵ + ( ℵ : M ) M . Proposition 3.10 If ℵ is a submodule of M and g : M → M ′ is an epimorphism such that Ker ( g ) ⊆ ℵ , then (i) g ( S − rad M PS . B . ( ℵ ) ) = S − rad M PS . B . g ( ℵ ) . (ii) g − 1 ( S − rad M PS . B . ( ℵ ′ ) ) = S − rad M PS . B . g − 1 ( ℵ ′ ) . Proof: (i) By the Definition (3.1) , we have S − rad M PS . B . ( ℵ ) = ∩ j ∈ J V j where V j is S-PS.B. submodule of M containing ℵ , j ∈ J and J be an index set. Therefore g ( S − rad M PS . B . ( ℵ ) ) = g ( ∩ j ∈ J V j ) and g ( S − rad M PS . B . ( ℵ ) ) = ∩ j ∈ J g ( V j ) with g ( ℵ ) ⊆ g ( V j ) , since Ker ( g ) ⊆ ℵ ⊆ V j . Thus g ( S − rad M PS . B . ( ℵ ) ) = S − rad M PS . B . g ( ℵ ) . (ii) Assume that ℵ ′ is a submodule of M ′ , then by Definition (3.1) , S − rad M PS . B . ( ℵ ′ ) = ∩ j ∈ J V j ′ the intersection is over all S-PS.B. submodules V j ′ of M ′ containing ℵ ′ . By Ref. 12 , we have g − 1 ( S − rad M PS . B . ( ℵ ′ ) ) = g − 1 ( ∩ j ∈ J V j ′ ) = ∩ j ∈ J g − 1 ( V j ′ ) the intersection is over all S-PS.B. submodules g − 1 ( V j ′ ) of M containing g − 1 ( ℵ ′ ) . Hence g − 1 ( S − rad M PS . B . ( ℵ ′ ) ) = S − rad M PS . B . g − 1 ( ℵ ′ ) . Proposition 3.11 If ℵ , K are two submodules of an F − module M , then S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) = ℵ ∩ K if and only if ℵ ∩ K is radical and S − rad M PS . B . ( ℵ ∩ K ) = S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) . Proof: Assume that S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) = ℵ ∩ K . From Proposition (3.4) part ( i ) , we have ℵ ∩ K ⊆ S − rad M PS . B . ( ℵ ∩ K ) , then S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) ⊆ S − rad M PS . B . ( ℵ ∩ K ) and from Proposition (3.5) the inequality S − rad M PS . B . ( ℵ ∩ K ) ⊆ S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) satisfies. Therefore, S − rad M PS . B . ( ℵ ∩ K ) = S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) . We get that ℵ ∩ K is S − rad M PS . B . submodule. Since S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) = ℵ ∩ K , thus we have the another result. The converse can be proved by the assumption ℵ ∩ K is radical so S − rad M PS . B . ( ℵ ∩ K ) = ℵ ∩ K and since S − rad M PS . B . ( ℵ ∩ K ) = S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) , we get S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) = ℵ ∩ K . Proposition 3.12 If ℵ , K are two submodules of a finitely generated multiplication F − module M such that ( ℵ : M ) and ( K : M ) are both radical ideals, then ( S − rad M PS . B . ( ℵ ∩ K ) : M ) = ( ℵ ∩ K : M ) . Proof: Clearly, ( ℵ ∩ K : M ) = ( ℵ : M ) ∩ ( K : M ) = ( ℵ : M ) ∩ ( K : M ) = ( ℵ ∩ K : M ) . Since M is finitely generated, then by Ref. 13 Theorem 4.4, we have ( ℵ ∩ K : M ) = ( S − rad M PS . B . ( ℵ ∩ K ) : M ) . Thus ( ℵ ∩ K : M ) = ( S − rad M PS . B . ( ℵ ∩ K ) : M ) . Corollary 3.13 If ℵ , K are two submodules of a multiplication finitely generated F − module M , then ( S − rad M PS . B . ( ℵ ∩ K ) : M ) = ( S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) : M ) . Proof: From Ref. 13 Theorem 4.4, we get ( S − rad M PS . B . ( ℵ ∩ K ) : M ) = ( ℵ ∩ K : M ) = ( ℵ : M ) ∩ ( K : M ) = ( S − rad M PS . B . ( ℵ ) : M ) ∩ ( S − rad M PS . B . ( K ) : M ) = ( S − rad M PS . B . ( ℵ ) ∩ S − rad M PS . B . ( K ) : M ) . Proposition 3.14 If ℵ is a submodule of a multiplication finitely generated F − module M , then S − rad M PS . B . ( ℵ ) = ( ℵ : M ) M . Proof: From, Ref. 14 , we have ( ℵ : M ) M ⊆ S − rad M PS . B . ( ℵ ) . Now, let L be an S-PS.B. submodule of M such that ℵ ⊆ L , then ( ℵ : M ) ⊆ ( L : M ) . Since M is a finitely generated multiplication module, then M is scalar 11 and hence ℵ is prime Lemma (2.4) and L is prime submodule so that ( L : M ) is prime ideal implies that ( ℵ : M ) ⊆ ( L : M ) . Therefore ( ℵ : M ) M ⊆ ( L : M ) M ⊆ L . Since L is an arbitrary S-PS.B. submodule containing ℵ , then S − rad M PS . B . ( ℵ ) ⊆ ( ℵ : M ) M . Thus S − rad M PS . B . ( ℵ ) = ( ℵ : M ) M . Proposition 3.15 If ℵ , K are two submodules of an F − module M such that ℵ ⊆ K ⊆ M , ℵ is a direct summand of M and rad( M ) be an essential submodule of M . If S − rad M PS . B . ( ℵ ) = S − rad M PS . B . ( K ) then ℵ = K . Proof: Suppose that ℵ is a direct summand of M , then ∃ ℵ ′ < M such that M = ℵ ⨁ ℵ ′ . Thus K = ℵ ⨁ ( K ∩ ℵ ′ ) implies that S − rad M PS . B . ( K ) = S − rad M PS . B . ( ℵ ) ⨁ S − rad M PS . B . ( K ∩ ℵ ′ ) . Therefore S − rad M PS . B . ( K ∩ ℵ ′ ) = 0 which can be written as rad( M ) ∩ ( K ∩ ℵ ′ ) = 0 and since rad( M ) is an essential submodule of M implies that ( K ∩ ℵ ′ ) = 0 , thus ℵ = K . Proposition 3.16 If ℵ , K are two submodules of a multiplication finitely generated F − module M , then ( ℵ : M ) + ( K : M ) = ( S − rad M PS . B . ( ℵ + K ) : M ) , where ℵ , K are both radical submodules of M . Proof: By, Ref. 15 , since M is finitely generated multiplication module, then we have ( I M : M ) = I + ( 0 : M ) for all ideal I of F . Consider the finitely generated F − module M / K and the ideal ( ℵ : M ) instead of M and I respectively, then ( ℵ : M ) + ( K : M ) = ( ℵ : M ) + ( 0 : M K ) = ( ( ℵ : M ) ( M K ) : M K ) = ( ( ( ℵ : M ) M + K K ) : M K ) = ( ( ℵ : M ) M + K : M ) = ( ℵ + K : M ) = ( S − rad M PS . B . ( ℵ + K ) : M ) . Corollary 3.17 Let ℵ be a submodule of a finitely generated multiplication F − module M . Then ( S − rad M PS . B . ( ( ℵ : M ) M ) : M ) = ( ℵ : M ) , where ℵ is radical submodules of M . Proof: Since ℵ is radical submodules of M , then ( ℵ : M ) M ⊆ ℵ implies that S − rad M PS . B . ( ( ℵ : M ) M ) ⊆ ℵ . Therefore ( S − rad M PS . B . ( ( ℵ : M ) M ) : M ) ⊆ ( ℵ : M ) . On the other hand ( ℵ : M ) ⊆ ( ( ℵ : M ) M : M ) ⊆ ( S − rad M PS . B . ( ( ℵ : M ) M ) : M ) . Therefore ( S − rad M PS . B . ( ( ℵ : M ) M ) : M ) = ( ℵ : M ) . Corollary 3.18 If M is a multiplication finitely generated F − module , then S − rad M PS . B . ( ( S − rad M PS . B . ( I M ) : M ) M ) = S − rad M PS . B . ( I M ) , where I is radical ideal of F . Proof: Since ( S − rad M PS . B . ( I M ) : M ) M ⊆ S − rad M PS . B . ( I M ) , then S − rad M PS . B . ( ( S − rad M PS . B . ( I M ) : M ) M ) ⊆ S − rad M PS . B . ( I M ) . On the other hand I M ⊆ S − rad M PS . B . ( I M ) implies that I ⊆ ( S − rad M PS . B . ( I M ) : M ) and thus I M ⊆ ( S − rad M PS . B . ( I M ) : M ) M which means S − rad M PS . B . ( I M ) ⊆ S − rad M PS . B . ( ( S − rad M PS . B . ( I M ) : M ) M ) . Therefore S − rad M PS . B . ( ( S − rad M PS . B . ( I M ) : M ) M ) = S − rad M PS . B . ( I M ) . Proposition 3.19 Let ℵ and K be two submodules of a scalar F − module M , then (i) If S − rad M PS . B . ( ℵ + K ) = M , then ∃ t ∈ F such that t M ⊆ S − rad M PS . B . ( ℵ ) and ( 1 − t ) M ⊆ S − rad M PS . B . ( K ) . (ii) If M is finitely generated module such that ( ℵ + K ) = M , then ∃ t ∈ F such that t M ⊆ ℵ and ( 1 − t ) M ⊆ K , where ℵ , K are radical submodules of M . Proof: (i) Since S − rad M PS . B . ( ℵ + K ) = M , then F = ( M : M ) = ( S − rad M PS . B . ( ℵ + K ) : M ) = ( S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) + S − rad M PS . B . ( K ) ) : M ) = ( S − rad M PS . B . ( ℵ ) : M ) + ( S − rad M PS . B . ( K ) : M ) . Therefore F = ( S − rad M PS . B . ( ℵ ) : M ) + ( S − rad M PS . B . ( K ) : M ) . Thus the required result is clear. (ii) Since ( ℵ + K ) = M = S − rad M PS . B . ( ℵ + K ) , by part ( i ) we have F = ( S − rad M PS . B . ( ℵ ) : M ) + ( S − rad M PS . B . ( K ) : M ) . By Ref. 16 , since M is finitely generated module we have F = ( ℵ : M ) + ( K : M ) and hence F = ( ℵ : M ) + ( K : M ) . Thus clearly result follows. Proposition 3.20 If ℵ 1 , ℵ 2 are two submodules of an F − module M and every S-PS.B. submodule L of M is completely irreducible such that ℵ 1 ∩ ℵ 2 ⊆ L , then S − rad M PS . B . ( ℵ 1 ∩ ℵ 2 ) = S − rad M PS . B . ( ℵ 1 ) ∩ S − rad M PS . B . ( ℵ 2 ) . Proof: By Proposition (3.5) , we have S − rad M PS . B . ( ℵ 1 ∩ ℵ 2 ) ⊆ S − rad M PS . B . ( ℵ 1 ) ∩ S − rad M PS . B . ( ℵ 2 ) . Let L be an S-PS.B. submodule of M such that ℵ 1 ∩ ℵ 2 ⊆ L . Since every S-PS.B. submodule of M that contains ℵ 1 ∩ ℵ 2 is completely irreducible, then either ℵ 1 ⊆ L or ℵ 2 ⊆ L . Thus S − rad M PS . B . ( ℵ 1 ) ⊆ L or S − rad M PS . B . ( ℵ 2 ) ⊆ L . Therefore S − rad M PS . B . ( ℵ 1 ) ∩ S − rad M PS . B . ( ℵ 2 ) ⊆ L for any L and S − rad M PS . B . ( ℵ 1 ∩ ℵ 2 ) . Therefore S − rad M PS . B . ( ℵ 1 ) ∩ S − rad M PS . B . ( ℵ 2 ) ⊆ S − rad M PS . B . ( ℵ 1 ∩ ℵ 2 ) . Hence S − rad M PS . B . ( ℵ 1 ∩ ℵ 2 ) = S − rad M PS . B . ( ℵ 1 ) ∩ S − rad M PS . B . ( ℵ 2 ) . Definition 3.21 Now, we give some notations: G S − PS . ( ℵ ) = { k : k ∈ Spec S ‐ PS . ( M ) , ℵ ⊆ k } , S − rad M PS . B . ( ℵ ) = ∩ k ∈ Spec S − PS . k , where Spec S − PS . denoted all S-PS.B. submodules of M . Proposition 3.22 Let M be an F − module. Then (i) G S − PS . ( ℵ 1 ) ∩ G S − PS . ( ℵ 2 ) = G S − PS . ( ℵ 1 + ℵ 2 ) . (ii) G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) ⊆ G S − PS . ( ℵ 1 ∩ ℵ 2 ) , where ℵ 1 , ℵ 2 are two submodules of M . Proof: (i) Since G S − PS . ( ℵ 1 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 1 ⊆ k } , G S − PS . ( ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 2 ⊆ k } , G S − PS . ( ℵ 1 ) ∩ G S − PS . ( ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 1 ⊆ k , ℵ 2 ⊆ k } , G S − PS . ( ℵ 1 + ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ( ℵ 1 + ℵ 2 ) ⊆ k } . Hence G S − PS . ( ℵ 1 ) ∩ G S − PS . ( ℵ 2 ) = G S − PS . ( ℵ 1 + ℵ 2 ) . (ii) Since G S − PS . ( ℵ 1 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 1 ⊆ k } , G S − PS . ( ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 2 ⊆ k } , G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 1 ⊆ k or ℵ 2 ⊆ k } . Also G S − PS . ( ℵ 1 ∩ ℵ 2 ) = { k : k ∈ Spec S − PS . ( M ) , ℵ 1 ∩ ℵ 2 ⊆ k } . Now, let L ∈ G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) then L is S-PS.B. submodule such that ℵ 1 ⊆ L or ℵ 2 ⊆ L . Thus ℵ 1 ∩ ℵ 2 ⊆ ℵ 1 ⊆ L and ℵ 1 ∩ ℵ 2 ⊆ ℵ 2 ⊆ L and hence L ∈ G S − PS . ( ℵ 1 ∩ ℵ 2 ) . Therefore G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) ⊆ G S − PS . ( ℵ 1 ∩ ℵ 2 ) . Corollary 3.23 If ℵ 1 , ℵ 2 are two submodules of an F − module M and every S-PS.B. submodule k of M is completely irreducible such that ℵ 1 ∩ ℵ 2 ⊆ k , then G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) = G S − PS . ( ℵ 1 ∩ ℵ 2 ) . Proof: It is sufficient to show G S − PS . ( ℵ 1 ∩ ℵ 2 ) ⊆ G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) . Let k ∈ G S − PS . ( ℵ 1 ∩ ℵ 2 ) implies that k is S-PS.B. submodule and ℵ 1 ∩ ℵ 2 ⊆ k . Since every S-PS.B. submodule of M that contains ℵ 1 ∩ ℵ 2 is completely irreducible, then either ℵ 1 ⊆ k or ℵ 2 ⊆ k which means k ∈ G S − PS . ( ℵ 1 ) or k ∈ G S − PS . ( ℵ 2 ) so that k ∈ G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) . Thus G S − PS . ( ℵ 1 ∩ ℵ 2 ) ⊆ G S − PS . ( ℵ 1 ) ∪ G S − PS . ( ℵ 2 ) . Corollary 3.24 Let M be a finitely generated multiplication F − module and I , J are ideals of F . Then (i) G S − PS . ( I M ) = G S − PS . ( I M ) . (ii) G S − PS . ( I M + J M ) = G S − PS . ( I M ) ∩ G S − PS . ( J M ) . (iii) G S − PS . ( ( I ∩ J ) M ) = G S − PS . ( I M ∩ J M ) = G S − PS . ( I M ) ∪ G S − PS . ( J M ) . Proof: (i) Assume that I is an ideal of F . Since I ⊆ I then I M ⊆ I M and hence G S − PS . ( I M ) ⊆ G S − PS . ( I M ) . Let ℵ ∈ G S − PS . ( I M ) , then ℵ is S-PS.B. submodule and since M is a finitely generated multiplication, then M is scalar 11 and thus ℵ is prime Lemma (2.4) . Since I ⊆ ( I M : M ) ⊆ ( ℵ : M ) , thus I ⊆ ( ℵ : M ) and hence ( I M : M ) ⊆ ( ( ℵ : M ) M : M ) ⊆ ( ℵ : M ) it follows that ℵ ∈ G S − PS . ( I M ) and thus G S − PS . ( I M ) ⊆ G S − PS . ( I M ) . (ii) Let I and J be two ideals of F . Suppose that ℵ ∈ G S − PS . ( I M ) ∩ G S − PS . ( J M ) , then ℵ is S-PS.B. submodule and since M is a finitely generated multiplication, then M is scalar 11 and thus ℵ is prime Lemma (2.4) . We have I ⊆ ( I M : M ) ⊆ ( ℵ : M ) and J ⊆ ( J M : M ) ⊆ ( ℵ : M ) . Thus ( I + J ) M ⊆ ( ℵ : M ) M ⊆ ℵ . Therefore, ( ( I + J ) M : M ) ⊆ ( ℵ : M ) and thus ℵ ∈ G S − PS . ( I M + J M ) . Clearly, G S − PS . ( I M + J M ) ⊆ G S − PS . ( I M ) ∩ G S − PS . ( J M ) . Therefore, G S − PS . ( I M + J M ) = G S − PS . ( I M ) ∩ G S − PS . ( J M ) . (iii) Let I and J be two ideals of F . Clearly, we have G S − PS . ( I M ) ∪ G S − PS . ( J M ) ⊆ G S − PS . ( I M ∩ J M ) ⊆ G S − PS . ( ( I ∩ J ) M ) . Let ℵ ∈ G S − PS . ( ( I ∩ J ) M ) , then ℵ is S-PS.B. submodule and from 11 and Lemma (2.4) hence ℵ is prime. So that I ∩ J ⊆ ( ( I ∩ J ) M : M ) ⊆ ( ℵ : M ) . Now, since ( ℵ : M ) is prime ideal of F , then I ⊆ ( ℵ : M ) or J ⊆ ( ℵ : M ) . Hence ( I M : M ) ⊆ ( ℵ : M ) or ⊆ ( J M : M ) ⊆ ( ℵ : M ) . It follows that ℵ ∈ G S − PS . ( I M ) or ℵ ∈ G S − PS . ( J M ) implies that ℵ ∈ G S − PS . ( I M ) ∪ G S − PS . ( J M ) . Corollary 3.25 If ℵ is a submodule of a multiplication finitely generated F − module M , then G S − PS . ( ℵ ) = G S − PS . ( ( ℵ : M ) M ) . Proof: Clearly, G S − PS . ( ℵ ) ⊆ G S − PS . ( ( ℵ : M ) M ) . Let L ∈ G S − PS . ( ( ℵ : M ) M ) , then we have ( ( ℵ : M ) M ) ⊆ L . Since ( ℵ : M ) ⊆ ( ( ℵ : M ) M : M ) ⊆ ( L : M ) so that ( ℵ : M ) ⊆ ( L : M ) and hence L ∈ G S − PS . ( ℵ ) . Thus G S − PS . ( ℵ ) = G S − PS . ( ( ℵ : M ) M ) . 4. Conclusion In this study, we have re-discovered and proved some properties of S-pseudo bounded radical submodule. Our results are supported by facts related to the prime submodule, prime ideal, and S-prime submodule. We examined the properties of the S-pseudo bounded radical submodule, particularly in the context of finitely generated and multiplication modules. In addition, using S-Pseudo bounded radical submodule 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. Reporting guidelines Thank you for the provided guidelines regarding reporting standards. I will ensure compliance with the relevant guidelines applicable to pure mathematics research to maintain clarity and scientific rigor. References 1. McCasland RL, Moore ME: On radicals of submodules. Communications in Algebra. 1991; 19 (5): 1327–1341. Publisher Full Text 2. Abdul-Al-Kalik AJ: Semi–Bounded Modules. Baghdad Sci. J. 2012; 9 (4): 20. Publisher Full Text 3. Al-Mothafar NS, Abdil-Khalik AJ: End-Ψ-Prime Submodules. Ibn Al-Haitham J. Pure Appl. Sci. 2016; 29 (2). Reference Source 4. Murad MS, Shihab BN: Endo-restricted bounded submodules related to prime submodules and scalar modules. AIP Conference Proceedings. AIP Publishing LLC; 2025; Vol. 3264 (1). Publisher Full Text 5. Al-Quraishi AD, Shihab BN: Pu-visible Submodule with their most prominent characteristics. IOP Conference Series: Materials Science and Engineering. IOP Publishing; 2020; Vol. 871 (1). Reference Source 6. Hadi IM, Ali SNA-A, Shyaa FD: T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules. Baghdad Sci. J. 2021; 18 (1): 0156. Publisher Full Text 7. Ramadhan HA, Al Mothafar NS: Weakly Small Semiprime Submodules. J. Phys. Conf. Ser. 2021; 1879 (3): 032128. IOP Publishing. Publisher Full Text 8. Abbas MR, Ahmed MA: Modules Whose St-Closed Submodules are Fully Invariant. Baghdad Sci. J. 2025; 22 (3): 988–996. Publisher Full Text 9. Ajeel AS, Mohammad HK, Ali.: Approximaitly Prime Submodules and Some Related Concepts. Ibn Al-Haitham J. Pure Appl. Sci. 2019; 32 (2): 103–113. Publisher Full Text 10. Hakeem A, Hashim MB, Al-Mothafar NS, et al. : Semi-Small Compressible Modules and Semi-Small Retractable Modules. Ibn Al-Haitham J. Pure Appl. Sci. 2023; 36 (4): 407–413. Publisher Full Text 11. Shihab BN: Scalar Reflexive Modules. Iraq: University of Baghdad; 2004. Doctoral dissertation, PhD. Thesis. Reference Source 12. Kasch F: Modules and rings. Vol. 17 . .Academic Press; 1982. Publisher Full Text 13. McCasland RL, Moore ME, Smith PF: Modules with bounded spectra. Commun. Algebra. 1998; 26 (10): 3403–3417. Publisher Full Text 14. McCasland RL, Moore ME: On radicals of submodules of finitely generated modules. Can. Math. Bull. 1986; 29 (1): 37–39. Publisher Full Text 15. Faith C: Algebra II Ring Theory: Vol. 2: Ring Theory. Vol. 191 . . Springer Science & Business Media; 2012. Publisher Full Text 16. McCasland RL, Moore ME: Prime submodules. Commun. Algebra. 1992; 20 (6): 1803–1817. 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: 17 Apr 2026, 14:1378 https://doi.org/10.12688/f1000research.172188.3 version 2 Revised Published: 20 Feb 2026, 14:1378 https://doi.org/10.12688/f1000research.172188.2 version 1 Published: 09 Dec 2025, 14:1378 https://doi.org/10.12688/f1000research.172188.1 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. S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.12688/f1000research.172188.2 ) 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 2 VERSION 2 PUBLISHED 20 Feb 2026 Revised Views 0 Cite How to cite this report: Abouhalaka A and Alhussein H. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.196524.r460752 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v2#referee-response-460752 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 12 Mar 2026 Alaa Abouhalaka , Çukurova University, Balcalı, Adana, Turkey Hassan Alhussein , Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Novosibirsk Oblast, Russian Federation Not Approved VIEWS 0 https://doi.org/10.5256/f1000research.196524.r460752 After carefully examining the revised version, the authors’ response to my previous report, and comparing the revised manuscript with the original one, I find that the changes are largely limited to minor editorial adjustments. The central definitions, ... Continue reading READ ALL After carefully examining the revised version, the authors’ response to my previous report, and comparing the revised manuscript with the original one, I find that the changes are largely limited to minor editorial adjustments. The central definitions, the main arguments, and the mathematical issues raised in my previous report remain essentially unchanged. Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebra-(Non)-commutative rings, Modules. We confirm that we have read this submission and believe that we have an appropriate level of expertise to state that we 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 Abouhalaka A and Alhussein H. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.196524.r460752 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v2#referee-response-460752 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 In the Abstract the third statement has been removed (the definition) Competing Interests: No competing interests In the Abstract the third statement has been removed (the definition) In the Abstract the third statement has been removed (the definition) 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 In the Abstract the third statement has been removed (the definition) Competing Interests: No competing interests In the Abstract the third statement has been removed (the definition) In the Abstract the third statement has been removed (the definition) Competing Interests: No competing interests Close Report a concern COMMENT ON THIS REPORT Version 1 VERSION 1 PUBLISHED 09 Dec 2025 Views 0 Cite How to cite this report: Oduor MO. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r441339 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v1#referee-response-441339 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 16 Jan 2026 Maurice Owino Oduor , Department of Mathematics, Actuarial and Physical Sciences, University of Kabianga, Kericho, Kericho County, Kenya Approved VIEWS 0 https://doi.org/10.5256/f1000research.189901.r441339 Title of Manuscript: S-Pseudo Bounded Radical of Submodules Overview The researchers have investigated some properties of S-pseudo Bounded Radicals of Submodules, particularly focusing on finitely generated and multiplication modules. They have advanced the theory of modules, contributing ... Continue reading READ ALL Title of Manuscript: S-Pseudo Bounded Radical of Submodules Overview The researchers have investigated some properties of S-pseudo Bounded Radicals of Submodules, particularly focusing on finitely generated and multiplication modules. They have advanced the theory of modules, contributing significantly to the knowledge of Abstract Algebra. The following are my observations. Abstract The authors should interchange the second and the third statements. The statement which begins with, “In our work many….” should be written in the past tense because the research has been done. The last sentence which begins with, “We used the symbol…”should be transferred to the Introduction Section. Introduction In the definition of use a vertical line to denote such that instead of a slash /. S-Pseudo bounded submodules The numbering of examples should follow a different order from Definitions and Theorems. For instance, Example 2.2 should read Example 1. In this example, replace the word ‘as’ with the word ‘be’. Replace the phrase ‘Then if we take’ with ‘If’. Delete the word ‘when’ appearing after ‘since’. Lemma 2.4. Proof. Replace the phrase ‘In the same way’ with ‘Similarly’ Some results related to S-pseudo bounded radical of submodules Delete the first three words. The statement to read as follows: We have examined the … Proposition 3.5. Proof. (i)Delete the word ‘that’ appearing after ‘Suppose’ (ii) The last sentence. Replace the word ‘easily’ with ‘easy’ Corollary 3.7. Proof. Merge the last sentence with the preceding section of the proof. There is unnecessary gap between the two. Proposition 3.10. Correct the spelling of the word ‘epimorphism’. Proposition 3.11. Proof. The last line of the proof which reads, ‘thus we have the another result’ is wrong. The proof of the converse is not clear. Proposition 3.12. Ref.11 in the proof is an inappropriate citation. Which result/theorem from [11] has been used? Corollary 3.13. Ref.11 as used in the proof is an inappropriate citation. Proposition 3.14. Ref.12 as used in the proof is an inappropriate citation. Proposition 3.16. Ref.13 as used in the proof an inappropriate citation. Proposition 3.19 to read: Let N and K be two submodules of a scalar F- module M . Then… The second part of Proposition 3.19, to read:…and , where …. are radical submodules of M . The proof of the first part of Proposition 3.19 is defective. It should not begin with ‘Since..’ Why is The notation defined below the proof of Proposition 3.20 should be transferred to Section 1 (Introduction). Further, replace the word ‘denoted’ with ‘denotes’. Corollary 3.23….and are ideals of . Then… Conclusion The word ‘several’ used in this section is inappropriate. To be deleted. The first statement to be restructured to read as follows: In this study, we have re-discovered and proved some properties of S-pseudo bounded radical submodules… There is need to recommend areas of future research. This will keep the research field active. References Some references are quite old. For instance, [1], [10], [11], [12], [14] may be replaced by more current references. General Observation What is/are the major result (s) of this study? It is important to state the major result (s) to reveal the significant contribution to knowledge in this area of study. 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? 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? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: Rings and Modules 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 Oduor MO. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r441339 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v1#referee-response-441339 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 20 Feb 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 20 Feb 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. As for the ancient sources, they comprise all the materials available to us that are relevant to the field and specific aspects of our research with our sincere appreciation. 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. As for the ancient sources, they comprise all the materials available to us that are relevant to the field and specific aspects of our research with our sincere appreciation. Competing Interests: No competing interests Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 20 Feb 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 20 Feb 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. As for the ancient sources, they comprise all the materials available to us that are relevant to the field and specific aspects of our research with our sincere appreciation. 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. As for the ancient sources, they comprise all the materials available to us that are relevant to the field and specific aspects of our research with our sincere appreciation. Competing Interests: No competing interests Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Abouhalaka A and Alhussein H. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r443903 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v1#referee-response-443903 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 Jan 2026 Alaa Abouhalaka , Çukurova University, Balcalı, Adana, Turkey Hassan Alhussein , Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Novosibirsk Oblast, Russian Federation Not Approved VIEWS 0 https://doi.org/10.5256/f1000research.189901.r443903 \subsection*{Summary of the Article} The manuscript introduces the concept of an ``S--pseudo bounded radical'' of submodules and tries to relate this notion to classical concepts such as prime submodules, multiplication modules, and scalar modules. The ... Continue reading READ ALL \subsection*{Summary of the Article} The manuscript introduces the concept of an ``S--pseudo bounded radical'' of submodules and tries to relate this notion to classical concepts such as prime submodules, multiplication modules, and scalar modules. The authors present a number of propositions and lemmas that aim to describe this new structure, together with some examples intended to support the results. \medskip \subsection*{Positive Aspects} In line with constructive peer review principles, we first acknowledge the following positive aspects of the paper: \begin{itemize} \item The topic belongs to an active research area in algebra, where generalizations of primeness conditions are widely studied. \item The authors make an effort to develop a new framework and to connect it with existing ideas in module and radical theory. \item The paper contains several results and examples, which shows that the authors have attempted to build a structured theory. \end{itemize} \medskip \subsection*{Major Concerns} After carefully reading the full manuscript, we find that the paper contains serious problems that affect the correctness, clarity, and reliability of the results. In accordance with the F1000Research review standards, our comments are specific and intended to help the authors improve the work. \begin{enumerate} \item \textbf{Inconsistency in the main definitions.} The definition of an S--pseudo bounded submodule in the abstract uses equality of annihilators, while Definition~2.1 in the main text uses equality of the \emph{radicals} of annihilators. These two conditions are not equivalent and lead to different mathematical meanings. The paper does not clarify which version is intended, yet later arguments rely on this unclear definition. A single, precise definition is necessary before the results can be evaluated. \item \textbf{The definition is logically unclear.} In Definition~2.1 it is not clear whether the condition should hold for \emph{some} $(\varphi,x)$ or for \emph{all} $(\varphi,x)$. However, in Lemma~2.4 the authors use the definition as if it holds for arbitrary $\varphi$ and $x$. This inconsistency makes the notion difficult to understand and weakens the validity of the results based on it. \item \textbf{Logical gaps in important proofs.} Several proofs contain serious logical problems. For example, in Lemma~2.4 the authors do not clearly explain why the element $\varphi(x)$ belongs to $N$. Another example is Remark~3.3. The authors claim that $S\text{-}\mathrm{rad}^{PS.B}_{M}(\aleph)$ is an $S$--PS.B.\ submodule, but they do not prove that the class of $S$--PS.B.\ submodules is closed under finite or arbitrary intersections. Since \[ S\text{-}\mathrm{rad}^{PS.B}_{M}(\aleph)=\bigcap_{j\in J}k_j, \] the key step (that such intersections remain $S$--PS.B.) is missing. The statement therefore requires a full and rigorous proof. By contrast, the proof of Proposition~3.4 is clear, correct, and easy to verify, yet it is presented with unnecessary detail. \item \textbf{Unclear use of the notion of $S$--prime submodules.} The manuscript uses the term ``$S$--prime submodule'' in Section~2 without a clear definition, and it is not always clear whether $S$ is a subset of the ring or of the endomorphism ring. This creates ambiguity and makes it hard to judge whether the statements are meaningful. For comparison, in the commutative case, if $S$ is a multiplicatively closed subset of a ring $R$, a proper submodule $P$ of an $R$--module $M$ with \[ (P:_R M)\cap S=\varnothing \] is called an \emph{$S$--prime submodule} if there exists $s\in S$ such that for all $a\in R$ and $m\in M$, \[ am\in P \ \Longrightarrow\ sa\in (P:_R M)\ \text{ or }\ sm\in P . \] This standard notion is different from the notion of ``S--prime'' used in the manuscript, which appears instead to be related to endomorphisms. \item \textbf{Notation, terminology, and exposition problems.} The notation is often inconsistent (for example, the ring is denoted by $F$ and $\mathcal{F}$ in different places), and some symbols are unfamiliar or unclear, such as $S\text{-}\mathrm{rad}^{\text{PS.B}}_{\mathcal{M}}(\aleph)$ and $\operatorname{Spec}^{S\text{-PS}}(M)$. The abbreviation ``$S$--PS.B.'' is used without being formally defined. Several definitions and statements are difficult to interpret, and the paper would benefit from major restructuring and clarification. These issues reduce readability and make the arguments hard to follow. \item \textbf{Language and formatting issues.} The manuscript contains many grammatical mistakes, typographical errors, and broken mathematical expressions. These presentation problems further reduce clarity. For example, at the end of Proposition~3.8 the paper states that \[ ``...\text {where } \sqrt{(N:M)} \text{ be a prime ideal of } F'', \] but it is not clear whether this is meant as an assumption or as a conclusion. No justification is given, and in general the claim does not follow from the previous arguments. The statement needs to be rewritten and properly proved. \end{enumerate} \medskip \subsection*{Overall Assessment and Recommendation} Following the F1000Research review guidelines, our assessment is based on the validity and correctness of the results rather than novelty. At the present stage, the paper contains serious definitional ambiguities and major problems in several proofs, which undermine confidence in the conclusions. For these reasons, we are unable to recommend the article for approval in its current form. \medskip \noindent\textbf{Decision: Not Approved.} \medskip We encourage the authors to reconsider the core definitions, provide complete and rigorous proofs, and substantially improve the exposition. With major revision and reformulation, the topic may eventually develop into a coherent contribution. 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? No 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 References 1. ŞENGELEN SEVİM E, ARABACI T, TEKİR Ü, KOÇ S: On S-prime submodules. TURKISH JOURNAL OF MATHEMATICS . 2019; 43 (2): 1036-1046 Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebra-(Non)-commutative rings, Modules. We confirm that we have read this submission and believe that we have an appropriate level of expertise to state that we 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 Abouhalaka A and Alhussein H. Reviewer Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r443903 ) The direct URL for this report is: https://f1000research.com/articles/14-1378/v1#referee-response-443903 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 20 Feb 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 20 Feb 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. S-pseudo bounded submodule is a newly introduced concept that has not been addressed in previous studies. We have presented its definition and fundamental properties in our earlier research papers. In addition, we established relationships linking our concept to classical algebraic notions like scalar, prime, S-prime and others also we proved that it satisfies closed under finite intersections, we have obtained previous results . On this basis, we present the current paper. 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. S-pseudo bounded submodule is a newly introduced concept that has not been addressed in previous studies. We have presented its definition and fundamental properties in our earlier research papers. In addition, we established relationships linking our concept to classical algebraic notions like scalar, prime, S-prime and others also we proved that it satisfies closed under finite intersections, we have obtained previous results . On this basis, we present the current paper. Competing Interests: No competing interests Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 20 Feb 2026 Amal Rashid , Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, 31001, Iraq 20 Feb 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. S-pseudo bounded submodule is a newly introduced concept that has not been addressed in previous studies. We have presented its definition and fundamental properties in our earlier research papers. In addition, we established relationships linking our concept to classical algebraic notions like scalar, prime, S-prime and others also we proved that it satisfies closed under finite intersections, we have obtained previous results . On this basis, we present the current paper. 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. S-pseudo bounded submodule is a newly introduced concept that has not been addressed in previous studies. We have presented its definition and fundamental properties in our earlier research papers. In addition, we established relationships linking our concept to classical algebraic notions like scalar, prime, S-prime and others also we proved that it satisfies closed under finite intersections, we have obtained previous results . On this basis, we present the current paper. 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 Version 3 (revision) 17 Apr 26 Version 2 (revision) 20 Feb 26 read Version 1 09 Dec 25 read read Alaa Abouhalaka , Çukurova University, Balcalı, Turkey Hassan Alhussein , Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Russian Federation Maurice Owino Oduor , University of Kabianga, Kericho, Kenya 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 Abouhalaka A 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. 12 Mar 2026 | for Version 2 Alaa Abouhalaka , Çukurova University, Balcalı, Adana, Turkey Hassan Alhussein , Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Novosibirsk Oblast, Russian Federation 0 Views copyright © 2026 Abouhalaka A 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 (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 After carefully examining the revised version, the authors’ response to my previous report, and comparing the revised manuscript with the original one, I find that the changes are largely limited to minor editorial adjustments. The central definitions, the main arguments, and the mathematical issues raised in my previous report remain essentially unchanged. Competing Interests No competing interests were disclosed. Reviewer Expertise Algebra-(Non)-commutative rings, Modules. We confirm that we have read this submission and believe that we have an appropriate level of expertise to state that we 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 In the Abstract the third statement has been removed (the definition) View more View less Competing Interests No competing interests reply Respond Report a concern Abouhalaka A and Alhussein H. Peer Review Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.196524.r460752) 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/14-1378/v2#referee-response-460752 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. 16 Jan 2026 | for Version 1 Maurice Owino Oduor , Department of Mathematics, Actuarial and Physical Sciences, 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 (1) 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 Title of Manuscript: S-Pseudo Bounded Radical of Submodules Overview The researchers have investigated some properties of S-pseudo Bounded Radicals of Submodules, particularly focusing on finitely generated and multiplication modules. They have advanced the theory of modules, contributing significantly to the knowledge of Abstract Algebra. The following are my observations. Abstract The authors should interchange the second and the third statements. The statement which begins with, “In our work many….” should be written in the past tense because the research has been done. The last sentence which begins with, “We used the symbol…”should be transferred to the Introduction Section. Introduction In the definition of use a vertical line to denote such that instead of a slash /. S-Pseudo bounded submodules The numbering of examples should follow a different order from Definitions and Theorems. For instance, Example 2.2 should read Example 1. In this example, replace the word ‘as’ with the word ‘be’. Replace the phrase ‘Then if we take’ with ‘If’. Delete the word ‘when’ appearing after ‘since’. Lemma 2.4. Proof. Replace the phrase ‘In the same way’ with ‘Similarly’ Some results related to S-pseudo bounded radical of submodules Delete the first three words. The statement to read as follows: We have examined the … Proposition 3.5. Proof. (i)Delete the word ‘that’ appearing after ‘Suppose’ (ii) The last sentence. Replace the word ‘easily’ with ‘easy’ Corollary 3.7. Proof. Merge the last sentence with the preceding section of the proof. There is unnecessary gap between the two. Proposition 3.10. Correct the spelling of the word ‘epimorphism’. Proposition 3.11. Proof. The last line of the proof which reads, ‘thus we have the another result’ is wrong. The proof of the converse is not clear. Proposition 3.12. Ref.11 in the proof is an inappropriate citation. Which result/theorem from [11] has been used? Corollary 3.13. Ref.11 as used in the proof is an inappropriate citation. Proposition 3.14. Ref.12 as used in the proof is an inappropriate citation. Proposition 3.16. Ref.13 as used in the proof an inappropriate citation. Proposition 3.19 to read: Let N and K be two submodules of a scalar F- module M . Then… The second part of Proposition 3.19, to read:…and , where …. are radical submodules of M . The proof of the first part of Proposition 3.19 is defective. It should not begin with ‘Since..’ Why is The notation defined below the proof of Proposition 3.20 should be transferred to Section 1 (Introduction). Further, replace the word ‘denoted’ with ‘denotes’. Corollary 3.23….and are ideals of . Then… Conclusion The word ‘several’ used in this section is inappropriate. To be deleted. The first statement to be restructured to read as follows: In this study, we have re-discovered and proved some properties of S-pseudo bounded radical submodules… There is need to recommend areas of future research. This will keep the research field active. References Some references are quite old. For instance, [1], [10], [11], [12], [14] may be replaced by more current references. General Observation What is/are the major result (s) of this study? It is important to state the major result (s) to reveal the significant contribution to knowledge in this area of study. 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? 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? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise Rings and Modules 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 (1) Author Response 20 Feb 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. As for the ancient sources, they comprise all the materials available to us that are relevant to the field and specific aspects of our research with our sincere appreciation. View more View less Competing Interests No competing interests reply Respond Report a concern Oduor MO. Peer Review Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r441339) 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/14-1378/v1#referee-response-441339 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Abouhalaka A 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. 07 Jan 2026 | for Version 1 Alaa Abouhalaka , Çukurova University, Balcalı, Adana, Turkey Hassan Alhussein , Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Novosibirsk Oblast, Russian Federation 0 Views copyright © 2026 Abouhalaka A 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 (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 \subsection*{Summary of the Article} The manuscript introduces the concept of an ``S--pseudo bounded radical'' of submodules and tries to relate this notion to classical concepts such as prime submodules, multiplication modules, and scalar modules. The authors present a number of propositions and lemmas that aim to describe this new structure, together with some examples intended to support the results. \medskip \subsection*{Positive Aspects} In line with constructive peer review principles, we first acknowledge the following positive aspects of the paper: \begin{itemize} \item The topic belongs to an active research area in algebra, where generalizations of primeness conditions are widely studied. \item The authors make an effort to develop a new framework and to connect it with existing ideas in module and radical theory. \item The paper contains several results and examples, which shows that the authors have attempted to build a structured theory. \end{itemize} \medskip \subsection*{Major Concerns} After carefully reading the full manuscript, we find that the paper contains serious problems that affect the correctness, clarity, and reliability of the results. In accordance with the F1000Research review standards, our comments are specific and intended to help the authors improve the work. \begin{enumerate} \item \textbf{Inconsistency in the main definitions.} The definition of an S--pseudo bounded submodule in the abstract uses equality of annihilators, while Definition~2.1 in the main text uses equality of the \emph{radicals} of annihilators. These two conditions are not equivalent and lead to different mathematical meanings. The paper does not clarify which version is intended, yet later arguments rely on this unclear definition. A single, precise definition is necessary before the results can be evaluated. \item \textbf{The definition is logically unclear.} In Definition~2.1 it is not clear whether the condition should hold for \emph{some} $(\varphi,x)$ or for \emph{all} $(\varphi,x)$. However, in Lemma~2.4 the authors use the definition as if it holds for arbitrary $\varphi$ and $x$. This inconsistency makes the notion difficult to understand and weakens the validity of the results based on it. \item \textbf{Logical gaps in important proofs.} Several proofs contain serious logical problems. For example, in Lemma~2.4 the authors do not clearly explain why the element $\varphi(x)$ belongs to $N$. Another example is Remark~3.3. The authors claim that $S\text{-}\mathrm{rad}^{PS.B}_{M}(\aleph)$ is an $S$--PS.B.\ submodule, but they do not prove that the class of $S$--PS.B.\ submodules is closed under finite or arbitrary intersections. Since \[ S\text{-}\mathrm{rad}^{PS.B}_{M}(\aleph)=\bigcap_{j\in J}k_j, \] the key step (that such intersections remain $S$--PS.B.) is missing. The statement therefore requires a full and rigorous proof. By contrast, the proof of Proposition~3.4 is clear, correct, and easy to verify, yet it is presented with unnecessary detail. \item \textbf{Unclear use of the notion of $S$--prime submodules.} The manuscript uses the term ``$S$--prime submodule'' in Section~2 without a clear definition, and it is not always clear whether $S$ is a subset of the ring or of the endomorphism ring. This creates ambiguity and makes it hard to judge whether the statements are meaningful. For comparison, in the commutative case, if $S$ is a multiplicatively closed subset of a ring $R$, a proper submodule $P$ of an $R$--module $M$ with \[ (P:_R M)\cap S=\varnothing \] is called an \emph{$S$--prime submodule} if there exists $s\in S$ such that for all $a\in R$ and $m\in M$, \[ am\in P \ \Longrightarrow\ sa\in (P:_R M)\ \text{ or }\ sm\in P . \] This standard notion is different from the notion of ``S--prime'' used in the manuscript, which appears instead to be related to endomorphisms. \item \textbf{Notation, terminology, and exposition problems.} The notation is often inconsistent (for example, the ring is denoted by $F$ and $\mathcal{F}$ in different places), and some symbols are unfamiliar or unclear, such as $S\text{-}\mathrm{rad}^{\text{PS.B}}_{\mathcal{M}}(\aleph)$ and $\operatorname{Spec}^{S\text{-PS}}(M)$. The abbreviation ``$S$--PS.B.'' is used without being formally defined. Several definitions and statements are difficult to interpret, and the paper would benefit from major restructuring and clarification. These issues reduce readability and make the arguments hard to follow. \item \textbf{Language and formatting issues.} The manuscript contains many grammatical mistakes, typographical errors, and broken mathematical expressions. These presentation problems further reduce clarity. For example, at the end of Proposition~3.8 the paper states that \[ ``...\text {where } \sqrt{(N:M)} \text{ be a prime ideal of } F'', \] but it is not clear whether this is meant as an assumption or as a conclusion. No justification is given, and in general the claim does not follow from the previous arguments. The statement needs to be rewritten and properly proved. \end{enumerate} \medskip \subsection*{Overall Assessment and Recommendation} Following the F1000Research review guidelines, our assessment is based on the validity and correctness of the results rather than novelty. At the present stage, the paper contains serious definitional ambiguities and major problems in several proofs, which undermine confidence in the conclusions. For these reasons, we are unable to recommend the article for approval in its current form. \medskip \noindent\textbf{Decision: Not Approved.} \medskip We encourage the authors to reconsider the core definitions, provide complete and rigorous proofs, and substantially improve the exposition. With major revision and reformulation, the topic may eventually develop into a coherent contribution. 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? No 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 References 1. ŞENGELEN SEVİM E, ARABACI T, TEKİR Ü, KOÇ S: On S-prime submodules. TURKISH JOURNAL OF MATHEMATICS . 2019; 43 (2): 1036-1046 Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Algebra-(Non)-commutative rings, Modules. We confirm that we have read this submission and believe that we have an appropriate level of expertise to state that we do not consider it to be of an acceptable scientific standard, for reasons outlined above. reply Respond to this report Responses (1) Author Response 20 Feb 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. S-pseudo bounded submodule is a newly introduced concept that has not been addressed in previous studies. We have presented its definition and fundamental properties in our earlier research papers. In addition, we established relationships linking our concept to classical algebraic notions like scalar, prime, S-prime and others also we proved that it satisfies closed under finite intersections, we have obtained previous results . On this basis, we present the current paper. View more View less Competing Interests No competing interests reply Respond Report a concern Abouhalaka A and Alhussein H. Peer Review Report For: S-Pseudo Bounded Radical of Submodules [version 2; peer review: 1 approved, 1 not approved] . F1000Research 2026, 14 :1378 ( https://doi.org/10.5256/f1000research.189901.r443903) 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/14-1378/v1#referee-response-443903 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. 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