398 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License The Radical of an Endo-Restricted Bounded Submodule Related to Prime Submodules Mohammed Salman Murad 1 , Buthyna Najad Shihab 2* 1,2 Department of Mathematics, College of Education for Pure Sciences, Ibn-AL-Haitham, University of Baghdad, Baghdad, Iraq . . *Corresponding Author. Received:10 February 2025 Accepted: 13 May 2025 Published: 20 October 2025 doi.org/10.30526/38.4.4130 Abstract In this paper, we present the concept of the radical of an Endo-Restricted Bounded submodule and establish its characterization, which is regarded as a new notion. In addition, we study the relationship between the radical of submodules and the radical of an Endo- Restricted Bounded submodules and this connection will give us important results in terms of their radical. Furthermore, this article will demonstrate numerous properties and corollaries that elucidate the concept of the Endo-Restricted Bounded submodule's radical. This work includes a new class of T-module as well as a T-submodule called the Endo-Restricted Bounded Module (submodule), written briefly as the Endo-R.B. module (submodule), provided with some examples that illustrate and clarify in a nice way this type of module (submodule). However, our focus will be on the radicals of the endo-R.B. submodule. Prime submodule and scalar module both play a crucial role in many properties that show the relationship between prime submodules and Endo-R.B submodules. Furthermore, some generalizations of prime submodules, such as S-prime submodules, are involved in this research. Keywords: Endo-R.B Radical submodule, Bounded module, S-Prime submodule, Scalar module. 1. Introduction The ring in this paper is commutative with identity denoted by T and is a unitary left-T- module. Motivated by the notion of bounded module, where a T-module is called bounded if there exists such that ( ) ( ) (1–3). A T-submodule A is called bounded if there exists an element such that ( ) ( ) (3). We introduced a new concept of module (submodule) namely an Endo-Restricted Bounded submodule (module) (Endo-R.B submodule (module)) and then we turned to the main purpose, which is the radical of an Endo-R.B submodule that will be defined later in this work with some important properties. An Endo-R.B submodule is a new type of T-submodule that has not been recognized previously by other authors, where A proper submodule A of a T-module is called Endo- R.B whenever ( ) , ( ) implies that ( ( )) ( ). We say that is an Endo-R.B T-module if every proper submodule is an Endo-R.B. Also, prime https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com https://orcid.org/0009-0005-7288-4292 mailto:mohammed.baqi2203p@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6553-2089 mailto:buthynashihab@gmail.com IHJPAS. 2025,38(4) 399 submodules play a crucial role in order to recall a radical of a submodule and its connection with the radical of an Endo-R.B submodule, where a submodule A of a T-module is said to be prime whenever , - (4–7). In addition, we found that if A is an Endo-R.B, then it is not necessary that A be a prime, and the following example shows that: Let = 〈 ̅〉 〈 ̅〉 ( ̅ ̅) ( ̅, ̅) ( ̅ ̅) , then A is an Endo-R.B. However, A is not a prime submodule since if ( ̅ ̅) ( ̅ ̅) but ( ̅ ̅) , - Also, we found another example that shows if a submodule A is prime ,then it is not necessary to be an Endo-R.B. Assume that = as a Z-module and 〈 ̅〉. Define as the same way in the previous example. The radical of a T-submodule denoted by ( ) and it is the intersection of all prime submodules of that contain N. In other word, ( ) * | +(8–11). In this paper, we prove many properties provided with some crucial conditions, and we will see the relationship between ( ) and the radical of Endo-R.B submodules. 2. Endo-R.B Submodules and Modules In this section, we give a brief introduction about Endo-R.B submodules as well as its modules with some examples. We refer to a proper submodule by the symbol and ( ) is the set of all endomorphisms of a T-module . 2.1.Definition: A proper T-submodule A of is said to be Endo-R.B if there exists an endomorphism of ( ( )) and ( ) for some such that ( ( )) ( ) We need just one an endomorphism of with an element belonging to a T-module defined in some way that satisfies two conditions, ( ) and ( ( )) ( ) Examples 1) Suppose that = Z, and let 〈 ̅〉 . Then we can find Z Z defined by ( ̅ ) ( ̅ ) . It is clear that is an endomorphism. Now, let ( ̅ ) ( ̅ ) ( ̅ ) , and hence we see that ( ̅ ) ( ) 〈 ̅〉 . We conclude that a submodule A is an Endo-R.B submodule of . 2) Assume = as a Z-module and A=〈 ̅〉. Define as ( ̅) ̅ ̅ and is an endomorphism. Then ( ̅) , but ( ) 〈 ̅〉 is not equal to ( ( ̅)) ( ̅) . Therefore, A is not the Endo-R.B submodule. Now, we are ready to give the definition of the Endo-R.B module with some examples that explain the structure of the definition. 2.2. Definition A T-module is called Endo-R.B module if every proper submodule of is an Endo-R.B submodule. Examples (1) as a Z-module is Endo-R.B where P is a prime number since ( ̅) is the only proper submodule of . To show that, take an endomorphism ( ) as such that ( ̅) ̅ ̅ then ( ̅) ( ̅) ( ̅) ( ̅) ( ̅) and (〈 ̅〉) ( ( ̅)) IHJPAS. 2025,38(4) 400 (2) Let as a Z-module. Define as ( ̅ ̅) ( ̅ ) ( ̅ ̅) . Then, if we take ( ̅) as a submodule of we get that ( ̅ ̅) ( ̅ ) . Thus, we conclude that is not an Endo-R.B Z-submodule because if ( ̅ ̅), then ( ) ( ̅ ̅) and, hence, is not an Endo-R.B Z-module. 2.3. Remark (1) Every Endo-R.B.T-module is bounded, but the converse is not true. To show that, let as a Z-module and let ( ̅) be a submodule of Define where ( ) by ( ̅ ̅) ( ̅ ̅) Since ( ̅ ̅) ( ̅ ̅) but ( ) ( ̅ ̅) . Therefore, is not an Endo-R.B, while is a bounded T-module since there exists an element ( ̅ ̅) such that ( ) ( ̅ ̅) . (2) Every proper submodule of the Endo-R.B module is also an Endo-R.B. (3) The intersection of two Endo-R,B submodules of is an Endo-R.B since if and are two Endo-R.B submodules. Then, which implies that is an Endo-R.B submodule. 2.4. Proposition Let be a T-module and . Then (1) If N is an Endo-R.B submodule of and an epimorphism ( ), then ( ) is an Endo-R.B submodule of (2) If N is an Endo-R.B submodule of , then ( ) is an Endo-R.B submodule of . Proof. (1) Define as ( ) and suppose , then ( ) because N is an Endo-R.B submodule of It is clear that ( ) we can define as ( ( )) . Then, we have the following commutative diagram: Since ( )( ) ( ( )) it is obvious that ( ( )) ( ). ( ( )) ( ) ( ) because is an epimorphism. Also, it is clear that ( ( )) ( ( ( )) Another containment, let ( ( (( )), then ( ( )) implies that ( ) for all , then and since is an epimorphism ( ( )) we obtain ( ) and hence, ( ( )) (2) Suppose that , then there exists defined as: ( ) and ( ) . Assume ( ) and define: ( ( )) ( ) Since is an isomorphism, then is onto and hence, ( ) From this fact, we get ( ( )) ( ) It is clear that: ( ( )) ( ( ( )) Now, let ( ( ( )) ( ( )). Thus, ( ) for all , then ( ) . Therefore, since is outomor Ω Ω 𝐼Ω Ω 𝜓 𝜑 𝜓 ∘ 𝜑 𝐼Ω IHJPAS. 2025,38(4) 401 Then ( ( ( )) ( ( )) The next proposition shows that a divisible T-module plays an important role for a cyclic submodule to be an Endo-R.B. 3. The radical of an Endo-R.B T-submodule In this section, we are ready to focus on the main purpose of this paper and give some definitions and properties that illustrate the notion of the radical of an Endo-R.B submodule. 3.1. Definition The radical of an Endo-R.B submodule N of a T-module is denoted by Endo- ( ) and defined as the intersection of all Endo-R.B submodules of that contains N. If there exists no Endo-R.B submodule of containing N, then we write: Endo- ( ) . If =T and N is an ideal of T, then Endo- ( ) is the intersection of all Endo-R.B ideals of T containing N. 3.2. Definition A proper submodule N of an T-module is called an Endo-R.B radical submodule if Endo- ( ) 3.3. Remark Let N be an Endo-R.B submodule of a T-module , then Endo- ( ) is also an Endo-R.B submodule. Proof. By the definition of the radical of an Endo-R.B submodule, we have Endo- ( ) *K| K is an Endo-R.B submodule and +. By using induction and remark (2.5), we conclude that Endo- ( ) is an Endo-R.B submodule. 3.4. Proposition Let be an epiomorphism and with . Then, (1) . ( )/ ( ) (2) . ( )/ ( )where Proof. (1) By the definition of the Endo-R.B radical of submodule, we have . ( )/ ( ) where K is an Endo-R.B submodule and . Since , then ( ( )) ( ), where ( ) ( ) and the intersection works over all Endo-R.B submodules ( ) Therefore, . ( )/ ( ) (2) Let , then . ( )/ where the intersection is over all Endo-R.B submodules D of with . Then ( ( ) ( ) ( ) where the intersection is over all Endo- R.B submodules ( ) of with ( ) ( ). Hence . ( )/ ( ) 3.5. Proposition Let be an T-module and . Then, the following statements hold: (1) ( ) (2) , then Endo- ( ) ( ) (3) Endo- . ( )/ ( ) (4) ( ) ( ) ( ) (5) ( ) ( ( ) ( )) IHJPAS. 2025,38(4) 402 Proof. (1) By the definition, we have that ( ) where the intersection runs over all Endo-R.B submodules K of with so that ( ). (2) Assume that and let K be an Endo-R.B submodule of with . Then implies that . ( ) ( ). (3) Since . ( )/ where the intersection is taken on all over Endo-R.B submodules W of with ( ) and from no (1), we get that ( ) Therefore, . ( )/ ( ) and by no.(1), we obtain ( ) ( ( )). (4) Let K be an Endo-R.B submodule of containing N and L. Since we have that ( ) . Thus, ( ) ( ). Similarly, we have ( ) ( ) Therefore, ( ) ( ) ( ). (5) Since ( ) ( ), then by (2) ( ) ( ( ) ( )) Now, let K be an Endo-R.B submodule of containing N+L. Since then and . Thus, ( ) ( ) . Therefore, . ( ) ( )/ ( ). Hence, ( ) ( ( ) ( )). Recall a T-module called a multiplication module if for every submodule A of there exists an ideal I of T such that .(12–14) Recall a T-module said to be a scalar module if for each ( ) there exists such that ( ) (15–17). Next, the scalar module plays a crucial role to connect prime and Endo-R.B submodules. We need the next proposition to see this relationship. 3.6. Proposition Let be a scalar T-module, and A is a prime submodule. Then A is an Endo-R.B. Proof. Let ( ) . Since is a scalar module, then for all ( ) there exists such that ( ) . Thus, ( ) . We claim that ( ) ( ) Let ( ), then ( ) ( ) ( ) for all Since A is a prime submodule, then either . We conclude that ( ). 3.7. Proposition Let be a multiplication finitely generated T-module and Then, if and only if ( ) ( ) . Proof. By proposition (3.5), we have that ( ) and ( ) so it IHJPAS. 2025,38(4) 403 is obvious that ( ) ( ) Conversely, assume that ( ) ( ) and let . Since is finitely generated, then there exists a maximal submodule K of such that . K is prime submodule and since is a multiplication T-module, then by (18), is a scalar T-module and hence, by proposition (3.6), we get that K is an Endo-R.B submodule. Therefore, ( ) and ( ) Thus ( ) ( ) implies that K= which is a contradiction. Hence N+L= . Recall a submodule N of a –module is said to be completely irreducible if for any two submodules of , implies that either or (19,20) 3.8. Proposition Let be a T-module and If every Endo-R.B submodules that contains is completely irreducible submodule, then ( ) ( ) ( ) Proof. It is clear that ( ) ( ) ( ). Let K be an Endo-R.B submodule such that Since K is completely irreducible, then either implies that ( ) or ( ) . Hence ( ) ( ) and this holds for any K and the intersection of all K is an ( ). Then either ( ) ( ) or ( ) ( ) . Therefore, ( ) ( ) ( ) Thus ( ) ( ) ( ) 3.9. Proposition Let be a scalar T-module and . Then ( ) ( ) Proof. Let K be a prime submodule that containing N, then by proposition (3.6), K is an Endo-R.B submodule containing N implies that ( ) . Hence, ( ) for all Endo-R.B submodules containing N. Therefore, ( ) ( ) 3.10. Proposition Let N and L be two submodules of a T-module , then ( ) ( ) if and only if 1. is a radical Endo-R.B submodule. 2. ( ) ( ) ( ). Proof. Suppose that ( ) ( ) then ( ) ( ) ( ) and the inequality ( ) ( ) ( ) holds by proposition (3.5). Now, by the assumption and proving part (2), we have is a radical Endo-R.B submodule. Conversely, it is obvious that (1) and (2) prove that: IHJPAS. 2025,38(4) 404 ( ) ( ) Recall a submodule A of a T-module called an S-prime if there exists ( ) such that ( ) implies that either ( ) (21–23). 3.11. Lemma Let be a scalar T-module and A is an Endo-R.B. submodule of . Then A is an S-prime submodule Proof Let ( ) and define as ( ) . Suppose that then, we have to prove that ( ) Since is a scalar and A is an Endo-R.B submodule, then ( ) which means that ( ) Therefore, A is an S-prime submodule. Note that every S-prime submodule is a prime submodule. 3.12. Lemma Let be a scalar T-module and Then, ( ) ( ). Proof. Since every S-prime submodule is a prime, then the proof is by lemma (3.11) and proposition (3.6). 3.13. Proposition Let N and L are two submodules of a multiplication finitely generated T-module such that , - , - are radical ideals, then , - , ( ) - Proof. Clearly , - , - , - √, - √, - √, - Since is finitely generated, then by theorem 4.4 in (24), we have √, - , ( ) - Using lemma (3.12), we conclude that , - , ( ) -. 3.14. Proposition Let N and L are two submodules of a multiplication finitely generated T-module . Then [Endo- ( ) - [Endo- ( ) Endo- ( ) - Proof. Since is a multiplication finitely generated module, then is a scalar module and hence by lemma (3.12), we have ( ) ( ). Thus, [Endo- ( ) - , ( ) - √, - √, - √, - [Endo- - [Endo- - , ( ) ( ) - 3.15. Proposition Let be a multiplication finitely generated T-module and . Then √, - ( ). Proof. Let K be an Endo-R.B submodule of containing N. Also, by lemma (3.11), if is a scalar and N is an Endo-R.B, then N is an S-prime and every S-prime is a prime submodule. IHJPAS. 2025,38(4) 405 Therefore, K is a prime submodule and , - is a prime ideal implies that , - , - and √, - , -. Hence √, - , - and since K is an arbitrary Endo- R.B submodule containing N so, we have that √, - ( ). Since is a multiplication finitely generated T-module then is a scalar module and using lemma (3.12), we have that ( ) ( ). By (24),we get that ( ) √, - Therefore, √, - ( ) ( ). Let N be a submodule of a T-module and Q be a multiplicative set of T, then ( ) * + is a submodule of contains N (25,26) and the closure of a submodule A is denoted by ( ) * , - + (27,28) 3.16. Proposition Let be a T-module and Then 1) ( ) ( ( )) where Q is a multiplicative set of T. 2) ( ) ( ( )). 3) ( ) (, -) for every ideal I of T. Proof. 1) Since N(Q) is a submodule of contains N, then by proposition (3.5), we have that ( ) ( ( )) 2) It is clear since ( ) is a submodule of contains N. 3) Since for ideal I of T, we have , -. Therefore, the result follows directly from proposition (3.5) Recall a submodule H of a T-module said to be fully invariant if ( ) for every ( ) (29). 3.17. Proposition Let N and L be two fully invariant submodules of a –module M and consider , then ( ) ( ) ( ). Proof. Since , then ( ) ( ) ( ). Thus, ( ) ( ) ( ). The set of all Endo-R.B submodules of a T-module is denoted by ( ). Consider the notation: ( ) * | ( ) + and so ( ) ⋂ ( ) . 3.18. Proposition Let be a T-module, then the following holds (1) ( ) ( ) and ( ) . (2) ( ) ( ) ( ) (3) ( ) ( ) ( ) for any fully invariant submodule N, L of . Proof. (1) and (2) are obvious. (3) Let N and L are two fully invariant submodules of , then by the definition, we have ( ) * | ( ) + ( ) * | ( ) + ( ) ( ) * | ( ) + ( ) * | ( ) + Now, take ( ) ( ), then is an Endo-R.B submodule such that IHJPAS. 2025,38(4) 406 Therefore, and and hence ( ). Recall the radical of a T-module denoted ( ) and it is the intersection of all maximal submodules of (30). 3.19. Proposition Suppose that ( ) and let where K is a direct summand of . Then ( ) ( ) if and only if K=L. Proof. Since K is a direct summand of , then there exists a submodule of such that . Hence, ( ) so that ( ) ( ) ( ). Therefore, ( ) and this can be written as ( ) ( ) implies that since rad( ) is an essential submodule of . Thus, K=L. 4. Conclusion We discussed in this paper the formula of the radical of an Endo-R.B submodule as a new type and proved that it is an Endo-R.B submodule of a T-module . Also, the relationship between prime and Endo-R.B submodules helps us to give many properties. 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