F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172188.1 Research Article Articles S-Pseudo Bounded Radical of Submodules

[version 1; peer review: 1 approved, 1 not approved]

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

No competing interests were disclosed.

 9 12 2025 2025 14 1378 1 12 2025 Copyright: © 2025 Madhi Rashid A and Najad Shihab B 2025 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In 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 .

 S-pseudo bounded radical Finitely generated module Prime submodule Multiplication modules Scalar module. The author(s) declared that no grants were involved in supporting this work. 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 denoted 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

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.

A proper submodule ℵ of an F − module M is said to be S-pseudo bounded submodule if there exists φ ∈ S = End ( M ) , x ∈ M such that φ ( x ) ∈ ℵ implies ann F ( ℵ ) = ann F φ ( x ) , End ( M ) the set of endomorphism maps over an F − module M .

Let M = ℤ 2 ⨁ ℤ 4 as a ℤ − module . Define φ : M ⟶ M as φ ( a ¯ , b ¯ ) = ( 0 , ¯ b ¯ ) , ∀ ( a ¯ , b ¯ ) ∈ M . Then if we take ℵ = ⟨ 0 ¯ ⟩ ⨁ ℤ 4 , then φ ( a ¯ , b ¯ ) ∈ ℵ . Hence ℵ is S-PS.B. ℤ − submodule of M , since when ( a ¯ , b ¯ ) = ( 0 ¯ , 2 ¯ ) ∈ M implies that ann ℤ ( ℵ ) = ann ℤ ( 0 ¯ , 2 ¯ ) = 2 ℤ .

Consider M = ℤ ⨁ ℤ 3 as a ℤ − module . Define φ : M ⟶ M as φ ( a , b ¯ ) = ( 0 , 0 ¯ ) , ∀ ( a , b ¯ ) ∈ M , then if we take 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 .

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.

If ℵ is a prime submodule of a scalar F − module M , then ℵ is S-PS.B. F − submodule .

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 . In a same way, ann F ℵ ⊆ ann F φ ( x ) . Hence, ℵ is S-PS.B. F − submodule .

If ℵ is S-PS.B. F − submodule of a scalar F − module M , then ℵ is S-prime submodule.

Assume that ℵ be a proper submodule of M and φ ( x ) ∈ ℵ , φ ∈ End ( M ) , x ∈ M . Let x ∉ ℵ , then we have to prove φ ( M ) ⊆ ℵ . Since ℵ is S-PS.B. F − submodule and M is scalar, then φ ( x ) ∈ ℵ , ∀ x ∈ M which means that φ ( M ) ⊆ ℵ .

If ℵ is a submodule of a scalar F − module M , then S − rad M PS . B . ( ℵ ) = rad ( ℵ ) .

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

In this part, we have examined the S-pseudo bounded radical of submodules presenting various properties exploring several relationships and introducing new results.

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 denoted 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 .

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 .

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.

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.

If ℵ , K are two submodules of an F − module M . Then

ℵ ⊆ S − rad M PS . B . ( ℵ ) .

If ℵ ⊆ K , then S − rad M PS . B . ( ℵ ) ⊆ S − rad M PS . B . ( K ) .

S − rad M PS . B . ( S − rad M PS . B . ( ℵ ) ) = S − rad M PS . B . ( ℵ ) .

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 ) .

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 ) .

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 ) ) .

If ℵ j is a submodule of an F − module M , J is an index set, j ∈ J , then

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 ) ) .

∑ 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 ) ) .

Suppose that 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.

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 easily 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 ) ) .

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 .

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 9 and thus K is S-PS.B. submodule Lemma (2.4). 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.

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 .

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 .

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 .

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 9 and thus ℵ is prime Lemma (2.5) 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 ) .

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 .

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 (2.6) which is contradiction. So ( ( ℵ + ( ℵ : M ) M ) : M ) = ( ℵ : M ) , therefore S − rad M PS . B . ( ℵ ) = ℵ + ( ℵ : M ) M .

If ℵ is a submodule of M and g : M → M ′ is an epiomorphism such that Ker ( g ) ⊆ ℵ , then

g ( S − rad M PS . B . ( ℵ ) ) = S − rad M PS . B . g ( ℵ ) .

g − 1 ( S − rad M PS . B . ( ℵ ′ ) ) = S − rad M PS . B . g − 1 ( ℵ ′ ) .

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 ) .

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. 10, 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 ′ ) .

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 ) .

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 together.

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 ) .

Clearly, ( ℵ ∩ K : M ) = ( ℵ : M ) ∩ ( K : M ) = ( ℵ : M ) ∩ ( K : M ) = ( ℵ ∩ K : M ) . Since M is finitely generated, then by Ref. 11 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 ) .

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 ) .

From Ref. 11, 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 ) .

If ℵ is a submodule of a multiplication finitely generated F − module M , then S − rad M PS . B . ( ℵ ) = ( ℵ : M ) M .

From, Ref. 12, 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 a finitely generated multiplication module, then M is scalar 9 and hence ℵ is prime Lemma( 2.5) 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 .

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 .

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 .

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 .

By, Ref. 13, 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 ) .

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 .

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 ) .

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 .

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 ) .

If ℵ , K are two submodules of a scalar F − module M , then

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 ) .

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 .

Since 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.

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. 14, since M is finitely generated module we have F = ( ℵ : M ) + ( K : M ) and hence F = ( ℵ : M ) + ( K : M ) . Thus clearly result follows.

Recall that 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 . 15, 16

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 ) .

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 ) .

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 .

Let M be an F − module. Then

G S − PS . ( ℵ 1 ) ∩ G S − PS . ( ℵ 2 ) = G S − PS . ( ℵ 1 + ℵ 2 ) .

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 .

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 ) .

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 ) .

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 ) .

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 ) .

Let M be a finitely generated multiplication F − module and I, J are an ideals of F . Then

G S − PS . ( I M ) = G S − PS . ( I M ) .

G S − PS . ( I M + J M ) = G S − PS . ( I M ) ∩ G S − PS . ( J M ) .

G S − PS . ( ( I ∩ J ) M ) = G S − PS . ( I M ∩ J M ) = G S − PS . ( I M ) ∪ G S − PS . ( J M ) .

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 9 and thus ℵ is prime Lemma (2.5). 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 ) .

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 9 and thus ℵ is prime Lemma (2.5). 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 ) .

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 9 and Lemma (2.5) 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 ) .

If ℵ is a submodule of a multiplication finitely generated F − module M , then G S − PS . ( ℵ ) = G S − PS . ( ( ℵ : M ) M ) .

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 presented several significant findings concerning the 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.

 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 McCasland RL Moore ME : On radicals of submodules. Communications in Algebra. 1991;19(5):1327–1341. 10.1080/00927879108824205 Abdul-Al-Kalik AJ : Semi–Bounded Modules. Baghdad Sci. J. 2012;9(4):20. 10.21123/bsj.2012.9.4.720-727 Al-Mothafar NS Abdil-Khalik AJ : End-Ψ-Prime Submodules. Ibn Al-Haitham J. Pure Appl. Sci. 2016;29(2). Reference Source 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). 10.1063/5.0258500 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 Hadi IM Ali SNA-A Shyaa FD : T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules. Baghdad Sci. J. 2021;18(1):0156. 10.21123/bsj.2021.18.1.0156 Ramadhan HA Al Mothafar NS : Weakly Small Semiprime Submodules. J. Phys. Conf. Ser. 2021;1879(3):032128. IOP Publishing. 10.1088/1742-6596/1879/3/032128 Abbas MR Ahmed MA : Modules Whose St-Closed Submodules are Fully Invariant. Baghdad Sci. J. 2025;22(3):988–996. 10.21123/bsj.2024.11434 Shihab BN : Scalar Reflexive Modules. Iraq: University of Baghdad;2004. Doctoral dissertation, PhD. Thesis. Reference Source Kasch F : Modules and rings. Vol.17. Academic Press;1982. 10.5282/ubm/epub.20922 McCasland RL Moore ME Smith PF : Modules with bounded spectra. Commun. Algebra. 1998;26(10):3403–3417. 10.1080/00927879808826348 McCasland RL Moore ME : On radicals of submodules of finitely generated modules. Can. Math. Bull. 1986;29(1):37–39. 10.4153/CMB-1986-006-7 Faith C : Algebra II Ring Theory: Vol. 2: Ring Theory. Vol.191. Springer Science & Business Media;2012. 10.1007/978-3-642-65321-6 McCasland RL Moore ME : Prime submodules. Commun. Algebra. 1992;20(6):1803–1817. 10.1080/00927879208824432 Ajeel AS Mohammad HK Ali. : Approximaitly Prime Submodules and Some Related Concepts. Ibn Al-Haitham J. Pure Appl. Sci. 2019;32(2):103–113. 10.30526/32.3.2288 Hakeem A Hashim MB Al-Mothafar NS : Semi-Small Compressible Modules and Semi-Small Retractable Modules. Ibn Al-Haitham J. Pure Appl. Sci. 2023;36(4):407–413. 10.30526/36.4.3156 10.5256/f1000research.189901.r441339 Reviewer response for version 1 Oduor Maurice Owino 1 Referee https://orcid.org/0009-0005-7725-5487 Department of Mathematics, Actuarial and Physical Sciences, University of Kabianga, Kericho, Kericho County, Kenya

Competing interests: No competing interests were disclosed.

 16 1 2026 Copyright: © 2026 Oduor MO 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

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

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

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

Yes

Are the conclusions drawn adequately supported by the results?

Partly

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

Yes

Reviewer Expertise:

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.

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

Competing interests: No competing interests

 11 2 2026

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

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.

 10.5256/f1000research.189901.r443903 Reviewer response for version 1 Abouhalaka Alaa 1 Referee https://orcid.org/0000-0002-5818-8064 Alhussein Hassan 2 Co-referee Çukurova University, Balcalı, Adana, Turkey Novosibirsk State University of Economics and Management (Ringgold ID: 334659), Novosibirsk, Novosibirsk Oblast, Russian Federation

Competing interests: No competing interests were disclosed.

 7 1 2026 Copyright: © 2026 Abouhalaka A and Alhussein H 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

\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

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

Not applicable

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

No source data required

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

Partly

Are the conclusions drawn adequately supported by the results?

Partly

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

No

Reviewer Expertise:

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.

References : On S-prime submodules. TURKISH JOURNAL OF MATHEMATICS .2019;43(2) : 10.3906/mat-1808-50 1036-1046 10.3906/mat-1808-50 Rashid Amal Mathematics, College of Education for Pure Sciences, Ibn-AL-Haitham, University of Baghdad, Heet, Anbar, Iraq

Competing interests: No competing interests

 11 2 2026

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

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.