301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Abstract In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule of an -module is called an approximaitly prime submodule of (for short app-prime submodule), if when ever , where , , implies that either or . So, an ideal of a ring is called app-prime ideal of if is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established. Keywords: Prime submodules, Approximaitly prime submodules, Approximaitly prime radical, Socle of submodules. 1. Introduction Throughout this article, we consider all rings as commutative rings with identity, and all modules as unital -modules. A proper submodule of an - module is prime, if whenever , for , then either or [ ]where [ ] . The class of prime submodules was introduced and systematically studied in 1978 by Dauns [1]. as a generalization of the class of prime ideals of rings and recently has received a good of attention from several authors see [2-8]. In this paper, we will recall some basic definitions. The socle of a module denoted by is the intersection of all essential submodules of [9]. where a non-zero submodule of an -module is called essential if for each non-zero submodule of [9]. An element in a module over integral domain is torsion element if for all [9]. The set of all torsion elements of denoted by is a submodule of . If then is said to be torsion free [9]. An - module is multiplication if each submodule is the form for some ideal of or [ ] [10]. A subset of a ring is called multiplicatively closed subset of if and for every [11]. If is a submodule of an -module , and is a multiplicatively closed subset of , then is a submodule of and Ali Sh. Ajeel Haibat K. Mohammadali Approximaitly Prime Submodules and Some Related Concepts Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Ali.shebl@st.tu.edu.iq Article history: Received 27 December 2018, Accepted 20 January 2019, Publish May 2019 10.30526/32.2.2148 Doi: Department of Mathematics, College of Computer Science and Mathematics, University of Tikrit, Tikrit, Iraq H.mohammadali@tu.edu.iq mailto:Ali.shebl@st.tu.edu.iq mailto:Ali.shebl@st.tu.edu.iq mailto:H.mohammadali@tu.edu.iq mailto:Ali.shebl@st.tu.edu.iq mailto:Ali.shebl@st.tu.edu.iq mailto:Ali.shebl@st.tu.edu.iq 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 [11]. A non-zero -module is compressible, if it is passable to embed in every non-zero submodule of [12]. 2. Approximaitly Prime Submodules In this section, we introduce the definition of approximaitly prime submodule as a generalization of a prime submodule, and give some basic properties, examples and characterizations of this concept. Definition (1) A proper submodule of an -module is called an approximaitly prime submodule of (for short app-prime submodule), if whenever , where , , implies that either or . So, an ideal of a ring is called app-prime ideal of if is an app-prime submodule of -module . The following results are characterizations of app-prime submodules. Proposition (2) Let be an -module, and be a submodule of . Then is an app-prime submodule of if and only if for every submodule of and every ideal of such that , implies that either or [ ]. Proof ⇒ Assume that , where is an ideal of , and is a submodule of , and suppose that , then there exists such that . Since , then for any , . But is an app-prime submodule of , and , hence [ ]. Thus [ ]. ⇐ Assume that , where , , then , so by hypothesis either [ ] or . That is either or [ ]. The following corollary is a consequence immediately of a Proposition (2). Corollary (3) Let be an -module, and be a submodule of . Then is an app-prime submodule of if and only if for every submodule of and any with , implies that either or [ ]. Remark (4) It is clear that every prime submodule of an -module is an app-prime submodule of , while the converse is not true as the following example shows that: Example (5) Consider the -module , and 〈 ̅〉, 〈 ̅〉.Therefore each , , if , then either 〈 ̅〉 〈 ̅〉 〈 ̅〉 or [〈 ̅〉 〈 ̅〉 ] . Thus is an app- prime submodule of , but is not prime submodule of , because ̅ , but neither ̅ nor [ ] . Proposition (6) Let be an -module, and be a submodule of , with . Then is a prime submodule of if and only if is an app-prime submodule of . 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Proof It is clear The following corollaries are direct consequence of proposition (2.6). Corollary (7) Let be an -module, and be a submodule of , with . Then is a prime submodule of if and only if is an app-prime submodule of . It is well-known that a torsion free -module has zero socle [13]. so we set the following result. Corollary (8) Let be a torsion free -module, and be a submodule of . Then is a prime submodule of if and only if is an app-prime submodule of . Proposition (9) Let be an app-prime submodule of an -module , with . Then [ ] is an app-prime ideal of . proof It is followed by proposition (6) and by [14, Prop. 2.8]. The convers of proposition (9) is not true in general, as the following example explain that. Example (10) Let , , and 〈 〉, then [ ] is an app-prime ideal in a ring . But is not an app-prime submodule of . Recall that an -module is called singular module provided . At the other extreme, we say that is non-singular module provided , where where the set of all essential right ideals of the ring , [9]. The following proposition shows that the converse of proposition (9) is true under certain conditions. Proposition (11) Let be a multiplication non-singular -module, and be a proper submodule of , with . Then is an app-prime submodule of if and only if [ ] is an app- prime ideal of . Proof ⇒ Follows by proposition (9). ⇐ Let , where , , then . But is multiplication, then for some ideal of . Thus , it follows that [ ]. Since [ ] is an app-prime ideal of , then either [ ] or [[ ] ] [ ] . Hence either [ ] or [ ] . But is a non- singular, then by [9]. we have . Thus, either or . Hence either or [ ]. Hence is an app-prime submodule of . Proposition (12) Let be a faithful multiplication -module, and be a proper submodule of , with . Then is an app-prime submodule of if and only if [ ] is an app-prime ideal of . 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Proof ⇒ Follows by proposition (9). ⇐ Let , where , , then . But is multiplication, then for some ideal of . Thus , it follows that [ ]. Since [ ] is an app-prime ideal of , then by corollary (3) either [ ] or [[ ] ] [ ] . Thus either [ ] or [ ] . Since is a faithful multiplication, then by [10,cor.2.14]. we have . It follows that either or . Hence either or [ ]. Hence is an app-prime submodule of . Proposition (13) Let be an -module, and be a submodule of such that [ ] is a maximal ideal of . Then is an app-prime submodule of . Proof Let , where , , with [ ]. Since [ ] is a maximal ideal of , then 〈 〉 [ ], where 〈 〉 is an ideal of generated by . Thus, there exists and [ ] such that , it follows that . Hence is an app-prime submodule of . Proposition (14) Let be an -module, and be a proper submodule of , with [ ] [ ], and proper submodule of for each submodule of such that [ ] is a prime ideal of . Then is an app-prime submodule of . Proof Assume that , where , , with . Then 〈 〉 and so [ ] [ ], then there exists [ ] and [ ]. That is and . That is implies that 〈 〉 . It follows that [ ]. But [ ] is a prime ideal of , then [ ]. Hence is an app-prime submodule of . Note It is well-know that if is a non-zero multiplication module, then [ ] [ ] for each submodule of with , where is a proper submodule of [14, Rem.2.15]. Now, we get the following corollary as a direct consequence of proposition (14). Corollary (15) Let be a multiplication -module, and be a proper submodule of , with[ ] is prime ideal of , and for each submodule of . Then is an app-prime submodule of . Proposition (16) Let be an -module, and are submodules of , with . If is an app-prime submodule of and , then is an app-prime submodule of . Proof Suppose that , where , . Since is an app-prime submodule of , then either or . But , implies that either or . Thus is an app-prime submodule of . 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Proposition (17) Let be an -module, and be a submodule of , with is an app-prime submodule of . Then is an app-prime submodule of . Proof Suppose that , where , . Hence , and so . But is an app-prime submodule of , then either or . Thus is an app-prime submodule of . Proposition (18) Let be an -module, and be a submodule of , with [ ] is a prime ideal of . Then for each multiplicatively closed subset of with [ ] if and only if is an app-prime submodule of . Proof ⇒ Assume that , where , , and suppose that and [ ]. Since is a multiplicatively closed subset of , then , and since [ ] is a prime ideal of , then it is clear that [ ] . But , implies that and hence which contradiction. Thus, either or [ ], therefore is an app-prime submodule of . ⇐ Let , then there exists such that . But is an app-prime submodule of , so either or [ ]. But [ ], implies that [ ] , which is a contradiction. Thus and hence . Proposition (19) Let be an -module, and be a maximal ideal of , with . Then is an app-prime submodule of . Proof Clearly, [ ].That is there exists [ ] and , then 〈 〉, where 〈 〉 is an ideal of generated by , thus there exist and such that . Hence for each . It follows that for each , hence , it follows that which is a contradiction. Then and hence [ ] , it follows that [ ] is a maximal ideal of , hence by proposition (13) is an app-prime submodule of . Proposition (20) Let be a faithful multiplication -module, and is an app-prime ideal of . Then is an app-prime submodule of . Proof Let , where , , then . But is multiplication, then for some ideal of . It follows that , and so . But is an app- prime ideal of , then by corollary (3) either or [ ] . Thus either or . But be a faithful multiplication module then by [10, cor. 2.14]. we have . Thus either 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 or . That is either [ ] or . Hence is an app-prime submodule of . Proposition (21) Let be a finitely generated multiplication non-singular -module, and is an app-prime ideal of , with . Then is an app-prime submodule of . Proof Let , where , , then . But is multiplication, then for some ideal of . It follows that , and so . But is an app- prime ideal of , then by corollary (3) either or [ ] . Thus either or . But is non-singular, then by [9]. we have . Thus either or . That is either [ ] or . Hence is an app-prime submodule of . Proposition (22) Let be an -module, and be a proper submodule of with [ ] [ ] for each submodule of such that and . then is an app-prime submodule of if and only if is a compressible -module. Proof ⇒ Assume that is an app-prime submodule of and be a submodule of with , therefore is a non-zero submodule of , we are going to emmbed inside . Since [ ] [ ], then there exists [ ] and [ ]. That is . Define → by for each . It is clear that is an -homomorphism. To prove that is one to one. Suppose that , then , so , that is . but is an app-prime submodule of , so either or . Since , it follows that either or . But , hence , so . Thus is monomorphism and is a compressible. ⇐ Suppose that is a compressible -module, and , where , , with that is since . Then 〈 〉 is a submodule of , hence there is a monomorphism → 〈 〉 , that is . Let , then 〈 〉 , then such that that is , implies that that is , it follows that for each , hence , that is . Thus is an app-prime submodule of . So, we get the following corollary as a direct consequence of proposition (22). Corollary (23) Let be a multiplication -module, and be a proper submodule of with . Then is an app-prime submodule of if and only if is a compressible -module. 301 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Proposition (24) Let be an -module, and be a proper submodule of such that [ ] [ ] for each submodule of with and .Then is an app-prime submodule of . Proof Suppose that , where , , with . Let 〈 〉, then 〈 〉, it follows that . And so [ 〈 〉] [ 〈 〉] [ ]. Hence [ ]. It follows that is an app-prime submodule of . Remark (25) The intersection of two app-prime submodules of an -module need not be an app- prime submodule of , as the following example shows that: Example (26) Let , , and . and are app-prime submodules of , but is not app-prime submodule of since but and [ ] Proposition (27) Let be an -module, are two app-prime submodules of with and . Then is an app-prime submodule of . Proof Suppose that , where , , then and . Since and are app-prime submodules of , so either or and either or . But and , then either or and either or , it follows that either or . Hence is an app-prime submodule of . Proposition (28) Let be an -module, are two submodules of with is not contained in and . If is an app-prime submodule of then is an app-prime submodule of . Proof Since is not contained in , then is a proper submodule of . Now, let , where , , then and . But is app-prime submodule of , then either or . But and , then we have either [ ] or [ ] . But , so by modular law we have either or . But by [13]. Coro. 9.9] we have , hence either or . Thus is an app-prime submodule of . Proposition (29) Let be an -module, and be a submodule of such that , where is a prime submodule of for each . Then is an app-prime submodule of . Proof Let , where , , then for each . Since is a prime submodule of for each , so either or [ ]. That is either 330 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 or , it follows that . Hence either or [ ]. Therefore is an app-prime submodule of . The following proposition shows that the invers image of app-prime submodule is app- prime. Proposition (30) Let be an -epimorphism, and be an app-prime submodule of . Then is an app-prime submodule of . Proof It is clear that is a proper submodule of . Now, suppose that , where , , then . But is an app-prime submodule of , implies that either or . If , then ( ) [15,Theo.(1.4)a]. That is . If and , then . That is ( ) [15,Theo.(1.4)a]. Hence . Thus is an app-prime submodule of . Proposition (31) Let be an -epimorphism, and be an app-prime submodule of with . Then is an app-prime submodule of . Proof is a proper submodule of . If not, that is , let , then , so there exists such that , that is , implies that , it follows that , hence contradiction. Now suppose that , where , . Since is an epimorphism, and , then there exists such that ,that is , so there exists such that , it follows that , so , then . but be an app-prime submodule of , then either or , and hence either or . But by [15,Theo.(1.4)a]. we have . So, we have either or . Hence is an app-prime submodule of . As a direct consequence of proposition (30) and proposition (31), we set the following result. Corollary (32) Let be an -module, are two submodules of with . Then is an app- prime submodule of if and only if is an app-prime submodule of . Proposition (33) Let be an -module, are two submodules of and is an app-prime submodule of with and [ ] [ ] then . Proof Since [ ] [ ], then there exists [ ] but [ ]. Let , so and , so , implies that . But ia an app- prime submodule of and [ ] then . Thus . 333 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 3. Approximaitly Prime Radical of Submodules In this section we introduce the notion of approximaitly prime radical of a submodule, and we establish several properties of this notion that are similarly to those of radical of submodules. Definition (34) Let b an -module, and is a submodule of .An app-prime radical of a submodule denoted by is defined as the intersection of all approximaitly prime submodules of which contain , if there exists no approximaitly prime submodule containing , we put . In the following proposition we introduce some basic properties of approximaitly prime radical. Proposition (35) Let be an -epimorphism, and is a submodule of wihe . Then ( ) . Proof Since where the intersection runs over all app-prime submodules of with , so ( ) . Since , then by [13, Lemm.(3.1.10) c]. ( ) where the intersection runs over all app- prime submodules of with . Thus ( ) . Proposition (36) Let be an -epimorphism, and is a submodule of . Then ( ) . Proof Since where the intersection runs over all app-prime submodules of with . Hence by [15,Lemm.(3.1.10)a]. ( ) where the intersection runs over all app-prime submodules of with . It follows that ( ) . Proposition (37) Let be an -module, and are two submodules of . Then: (1) . (2) If , then . (3) ( ) . (4) . (5) ( ). Proof (1) Since where the intersection runs over all app-prime submodules of with , so . (2) Suppose that , and let be an app-prime submodule of with , then , implies that . Thus . (3) By part (1) we have ( ). But ( ) , where the intersection runs over all app-prime submodules of with , again by (1) . Thus 331 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 ( ) . Hence ( ) . (4) Let be an app-prime submodule of contining and . Since , so : . Thus . By same way . Hence . (5) Since , then by (2) we have ( ). Now, let be an app-prime submodule of with , we prove that . Since and , , then and . Hence . Therefore and we have ( ) . Recall that a submodule of an -module is called completely irreducible, if for any submodules of , , implies that either or [11]. Proposition (38) Let and are two submodules of an -module . If every app-prime submodule of which contains is a completely irreducible submodule, then . Proof holds by proposition (37)(4). Now, if , then . If , then there exists an app-prime submodule of such that or , so that either or because every app-prime submodule containing is completely irreducible, then we have either or . Therefore , and hence . 4. Conclusion In this paper an approximaitly prime submodules are introduced and studied as a new generalization of prime submodules, also we introduced and studied the approximaitly prime radical of modules. The main results of this study are the following. 1) A proper submodule of an -module is an app-prime submodule if and only , with is an ideal of and is a submodule of , implies that either or [ ]. 2) Every prime submodule is an app-prime submodule, while the converse is not true see Remark (4) and Example (5). 3) In multiplication non-singular -module a proper submodule of with is an app-prime submodule of if and only if [ ] is an app-prime ideal of . Also, this result is satisfied if is faithful multiplication with see proposition (12). 4) If is a proper submodule of , with [ ] is a prime ideal of . Then with [ ] if and only if is an app-prime submodule of . 331 Ibn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 5) If is a maximal ideal of , with . Then is an app-prime submodule This result is true if faithful multiplication (finitely generated multiplication non-singular) see proposition (20), proposition (21). 6) If [ ] [ ] for each submodule of with and .Then is an app-prime. 7) The invers image and homomorphic image of an app-prime submodule is an app-prime submodule see proposition (30), proposition (31). 8) We introduced and studied and state several basic properties of this notion for example see proposition (36), proposition (37) and proposition (38). References 1. Dauns, J. Prime Modules. J. reine Angew, Math.1978, 2, 156-181. 2. Lu, C.P. Prime Submodules of Modules. comm. Math. Univ. Sancti Pauli.1984, 33, 61- 69. 3. 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