48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Pseudo Weakly Closed Submodules and Related Concepts Haibat K. Mohammadali Mohammad E. Dahash Department of Mathematics College of Computer Science and Mathematics Tikrit University Abstract Let be a commutative ring with identity, and be a unitary left -module. In this paper we introduce the concept pseudo weakly closed submodule as a generalization of -closed submodules, where a submodule of an -module is called a pseudo weakly closed submodule, if for all , there exists a -closed submodule of with is a submodule of such that . Several basic properties, examples and results of pseudo weakly closed submodules are given. Furthermore the behavior of pseudo weakly closed submodules in class of multiplication modules are studied. On the other hand modules with chain conditions on pseudo weakly closed submodules are established. Also, the relationships of pseudo weakly closed submodules with other classes of modules are discussed. Keywords: Closed submodules, -closed submodules, pseudo weakly closed submodules, semi-prime submodules, fully semi-prime submodules, weakly essential submodules. 1. Introduction A proper submodule of an -module is called closed in , provided that has no proper essential extensions in [ ]. Where a non-zero submodule of an -module is called essential in if for each non-zero submodule of [ ]. And a non- zero submodule of is called weakly essential submodule of if for each non-zero semi-prime submodule of [ ]. Equivalently is weak essential, if whenever , then for every semi-prime submodule of [ ]. Where a submodule of is called semi-prime if whenever , for , , implies that [ ]. The concept of closed submodule recently extended by [ ] To -closed submodule, where a submodule of is called -closed submodule of if has no proper weak-essential extensions in , that is if is a weak essential submodule of , where is a submodule of , then [6,7]. This concept is generalized in this article to a pseudo weakly closed submodule. Many basic properties of this concept are discussed. Finally, we notes that throughout this paper all rings are commutative with identity and all modules Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.1.2376 dr.mohammadali2013@gmail.com Article history: Received 11 April 2019, Accepted 3 July 2019, Publish January 2020. mailto:dr.mohammadali2013@gmail.com 48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 are unitary left -modules, unless otherwise. Also, in this paper all -module under study contains semi-prime submodules. 2. Pseudo Weakly Closed Submodules In this section we introduce the notion of pseudo weakly closed submodule as a generalization of -closed submodule and give some basic properties and examples of this class. Definition 2.1 A submodule of an -module is called pseudo weakly closed submodule( for a short -closed), if for each , there exists a -closed of with such that . An ideal of a ring is called -closed if it is -closed submodule of an - module . Remarks and Examples 2.2 1. It is clear that every -closed submodule of an -module is -closed submodule of , but the converse is not true in general as the following example explain that: Consider the -module . The proper submodules of are: 〈 ̅〉, 〈 ̅〉, 〈 ̅〉 and 〈 ̅〉. The submodule { ̅ ̅ ̅} is -closed but not closed because ̅, ̅, ̅, ̅, ̅ ̅̅̅̅ are in but not in , then there exists a -closed submodule 〈 ̅〉 in such that and ̅, ̅, ̅, ̅, ̅ ̅̅̅̅ are not in . Now is not a -closed since is a weak essential submodule of . 2. A direct summand of an -module is not necessary -closed submodule in , for example: Consider the -module . In this module there are ten proper submodules 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅̅̅̅ 〉, 〈 ̅̅̅̅ 〉, 〈 ̅̅̅̅ 〉, 〈 ̅̅̅̅ 〉, 〈 ̅̅̅̅ 〉 where 〈 ̅〉 〈 ̅̅̅̅ 〉 . That is both and are a direct summands in . But 〈 ̅〉 is not -closed submodule in , since ̅ , ̅ 〈 ̅〉, then there exists a -closed submodule 〈 ̅〉 in with 〈 ̅〉 〈 ̅〉, but ̅ 〈 ̅〉. Also 〈 ̅̅̅̅ 〉 is not -closed submodule in , since ̅ , ̅ 〈 ̅̅̅̅ 〉, then there exists a -closed submodule 〈 ̅〉 in such that 〈 ̅̅̅̅ 〉 〈 ̅〉, but ̅ 〈 ̅〉. 3. If is a -closed submodule in an -module , then [ ] need not be a -closed ideal in . For example: Consider the -module . In this module there are four proper submodules, which are: 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅̅̅̅ 〉. Not that the submodule is a -closed submodule in ( since is a -closed submodule in and hence by (1) is a - closed ). While [ ] is not a -closed ideal in . Since there is no a -closed ideal in containing . 4. If is a module and is a -closed submodule in with is a submodule of such that , then it is not necessary that is a -closed submodule in . For example: The -module is a -closed submodule in ( since as a -module is a -closed in ) and , but is not a -closed submodule in . 5. If is an -module, and are submodules of an -module such that and is a -closed submodule in , then need not to be a -closed submodule in . For example: 48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Consider the -module is a -closed submodule in and , while is not - closed submodule in , , since there is no a -closed submodule in containing . 6. If is an -module, and are submodules of an -module such that and is a -closed submodule in , then need not to be a -closed submodule in . For example: Consider the -module and the submodules and . Notes that is a - closed submodule in ( since is a -closed in ), but is not -closed submodule in , since there is no a -closed submodule in containing . 7. [ ] shows that the intersection of two closed submodules need not to be closed submodule”. Also, the intersection of two -closed submodules need not to be - closed submodule as the following example shows: The -module the submodules 〈 ̅ ̅ 〉 and 〈 ̅ ̅ 〉 are -closed submodules in ( since they are -closed submodules in ), but 〈 ̅ ̅ 〉 〈 ̅ ̅ 〉 〈 ̅ ̅ 〉 is not - closed submodule in since 〈 ̅ ̅ 〉 is closed submodule in . 8. Closed submodules and -closed submodules are independent concepts, as the following examples show that: Consider the -module . In this module the proper submodules of are: 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅〉, 〈 ̅̅̅̅ 〉, 〈 ̅̅̅̅ 〉. We notes that the submodule 〈 ̅〉 is closed in , since has no proper essential extension in , while 〈 ̅〉 is not -closed submodule in since ̅ , ̅ , then there exists a -closed submodule 〈 ̅〉 in with , but ̅ . That is 〈 ̅〉 closed submodule in , but not -closed submodule in . In the -module , the submodule 〈 ̅〉 is a - closed submodule in since〈 ̅〉 is a -closed in , while 〈 ̅〉 is not closed in , since 〈 ̅〉 is an essential submodule of . We start this section by the following proposition. Proposition 2.3 If is an -module, and are submodules of with and is a -closed submodule in and is a -closed submodule in , then is a -closed submodule in , provided that for any weak-essential extensions of . Proof: To prove that is a -closed submodule in , suppose that with , then either or . If and since is a -closed submodule in , so there exists a -closed submodule in such that and . Since is a -closed in and is a -closed in , then by [ ] is a -closed in . Thus we have a -closed submodule in such that and . i.e. is a -closed submodule in . If , then nothing to prove since is a -closed submodule in such that and . Therefore is a -closed submodule in . Proposition 2.4 Let and be submodules of a module with . If is a -closed submodule in , and is a -closed in , then is a -closed submodule in , provided that is for any weak essential extensions of . 48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proof: Similar as in proposition 2.3. Proposition 2.5 If is a uniserial -module, and are submodules of such that and is a -closed submodule in and is a -closed submodule in , then is a -closed submodule in . Proof: Prove is direct. Recall that an -module is completely essential if every non-zero weak essential submodule of is essential [5, 3]. As we mention in Example and Remarks (2.2)(8) closed submodules and -closed submodules are independent, then the following propositions show that, the class of closed submodules is contained in the class of a -closed submodules under certain condition. Proposition 2.6 Let be a non zero closed submodule of a module such that every weak essential extension of is a completely essential submodule of , then is a -closed submodule in . Proof Assume that be a non zero closed submodule of , then by [ ] we get is a -closed submodule in , so by remarks and examples ( 2.2 )(1) we get is a -closed submodule of . Recall that an -module is called fully semi-prime, if every proper submodule of is a semi-prime submodule [ ]. Proposition 2.7 Let be a non zero closed submodule of a fully semi-prime -module . Then is a -closed submodule in . Proof Suppose that is a non zero closed submodule of , then by [ ] we get is a -closed submodule in . Hence by remarks and examples ( 2.2 )(1) is a -closed submodule in . Proposition 2.8 If and are submodules of an -module , with , containing every -closed submodule of and is a -closed submodule in , then is a a -closed submodule in . Proof Let and , then . Since is a -closed submodule in , then a -closed submodule in with and . Hence is a -closed submodule in with and . Thus is a a -closed submodule in . 44 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proposition 2.9 If is a module where and are submodules of , provided that and each weak essential extension of is completely essential module, where is a non-zero -closed submodule in and is a non-zero -closed submodule in , then is a -closed submodule in . Proof Let with , then either or . If and since is a -closed submodule in , so there exists a -closed submodule in with and . By [ ] is a -closed in , then by [ ], we have is a -closed in such that and . Similarly if , then a -closed submodule in containing and does not contain . Thus is a -closed submodule in . Proposition 2.10 If is an -module where and are submodules of , provided that and all submodules of are completely essential submodules. If is a non-zero -closed submodule in and is a non-zero -closed submodule in , then is a -closed submodule in if and only if is a -closed submodule in and is a -closed submodule in . Proof To prove is a -closed submodule in . If with . Then . But is a -closed submodule in , so there exists a -closed submodule in with and . Since , then by [ ] any submodule of can be written as , where and . Hence by [ ] it follows that is a -closed submodule in and is a - closed submodule in . Since and , then . Therefore is a - closed submodule in . Similarly is a -closed submodule in . Suppose that is a -closed submodule in and is a -closed submodule in . Let with , then either or . If and is a -closed submodule in , so a -closed submodule in such that and . But is a -closed submodule in and by [ ] is a -closed in , hence by [ ], we have is a -closed submodule in . Also and . Similarly if , then there exists a -closed submodule in containing and does not containing . Thus is a -closed submodule in . It is well known that a fully semi-prime module is a completely essential [ ] so we get the following corollary. Corollary 2.11 If is an -module where and are submodules of , with and all submodules of are fully semi-prime and is a non-zero -closed submodule in and is a non-zero -closed submodule in such that is a - 48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 closed submodule in , then is a -closed submodule in and is a -closed submodule in . Proposition 2.12 If is an epimorphism and let be a submodule of such that and is a -closed submodule in , then is a -closed submodule in , where is the intersection of all semi-prime submodule of . Proof Suppose that is a -closed submodule in and with . Since is an epi-morphism then there exists with and . Since is a - closed submodule in , then there exists a -closed submodule in with and . Thus by [ ] is a -closed submodule in . Since is an epi- morphism, then and . Thus is a -closed submodule in . Corollary 2.13 If and are submodules of a module with and is a -closed submodule in , then is a -closed submodule in . Proof It is clear. Recall that a submodule of an -module is called -closed submodule of , if is a non-singular module [ ]. The following Proposition gives a relationships between -closed submodule of and -closed submodule of . Proposition 2.14 If is a fully semi-prime module, then every non- zero -closed submodule is a - closed submodule in . Proof Suppose that is a non-zero -closed submodule of , then by [ ] is a - closed submodule in . Hence by remarks and examples ( 2.2 )(1) is a -closed submodule in . Proposition 2.15 If is an -module, and are non-zero submodules of with and every weak essential extension of is a completely essential submodule of such that is a -closed submodule in and is a -closed submodule in , then is a -closed submodule in . Proof Assume that with , then either with . If and since is a -closed submodule in , then a -closed submodule in with and . Since is a -closed submodule in and is a -closed submodule in , then by [ ], we get is a -closed submodule in . Thus is a -closed submodule in . If and is a -closed submodule in such that and . Thus is a -closed submodule in . 89 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proposition 2.16 If is a fully semi-prime -module and be a non-zero -closed submodule in and is a -closed submodule in , then is a -closed submodule in . Proof By using [ ] and similarly as in proposition (2.15), we get the result. Proposition 2.17 Let is a fully semi-prime module, and are submodules of , if is a -closed submodule in and is a weak essential submodule in . Then is a -closed submodule in . Proof Suppose that with implies that . Since , then . Since is a -closed submodule in , so there exists a -closed submodule in with and . Hence is a closed submodule in . And by [ ], is essential in . Therefore by [ ] we get is a closed submodule in . Hence by [ ] we have is a -closed submodule in with and . Hence is a -closed submodule in . Recall that an -module is called fully prime, if every proper submodule of is a prime submodule [ ]. It is well- known every fully prime -module is a fully semi-prime we get the following result. Corollary 2.18 Let be a fully prime module, and are submodules of , if is a -closed submodule in and is a weak essential submodule in . Then is a -closed submodule in . Since -closed submodule is a -closed submodule, then we get the following result. Corollary 2.19 Let be a fully semi-prime module, and are submodules of , if is a -closed submodule in and is a weak essential submodule in . Then is a -closed submodule in . Since in the class of a fully semi-prime modules, -closed submodule is a -closed submodule we get the following result. Corollary 2.20 If is a fully semi-prime module, and are submodules of , if is a -closed submodule in and is a weak essential submodule in . Then is a -closed submodule in . 3. -Closed Submodules in Multiplication Modules This section is devoted to study the behavior of -closed submodules in the class of multiplication modules. Recall that an -module is a multiplication if every submodule of is of the form for some ideal of [ ]”. And an -module is called faithful if for any non-zero , there is an element such that [ ]. 89 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Recall that for any -module and any ideals and of , if is a semi-prime ideal of , then is a semi-prime submodule of . This is called condition(*) [ ]. Proposition 3.1 If is a faithful multiplication module satisfies condition(*), and are ideals of a ring such that is a -closed ideal in , then is a -closed submodule in . Proof Let be a -closed ideal in , and let where , and with i.e. , but . But is a -closed ideal in and , then there exists a -closed ideal in such that and . But is a faithful and multiplication then and . But by [ ], is a -closed submodule in . Hence is a -closed submodule in . Proposition 3.2 If is a finitely generated, faithful and multiplication - module, and are ideals of a ring such that is a -closed submodule in , then is a -closed ideal in . Proof Suppose that is a -closed submodule in . Let with , there for each , and . But is a -closed submodule in then there exists a -closed submodule in , where is an ideal in such that and . Hence by [ ] we get is a -closed ideal in , and with and . Hence is a -closed ideal in . From proposition ( 3.1 ) and proposition ( 3.2 ), we get the following corollaries Corollary 3.3 Let be a finitely generated, faithful and multiplication - module which satisfies condition (*). Then is a -closed ideal in if and only if is a -closed submodule in . Corollary 3.4 Let be a finitely generated multiplication module and let be a submodule of such that satisfies condition (*). Then the next statements are equivalents. 1. is a -closed submodule in . 2. [ ] is a -closed ideal in . 3. for some -closed ideal in . Proof (1) (2) Since is a multiplication module and is a -closed submodule in , then by [ ] [ ] , hence by proposition ( 3.1 ) [ ] is a -closed ideal in . (2) (3) Following by [ ] and by proposition ( 3.2 ). (3) (1) Since is a -closed ideal in , then by proposition ( 3.2 ) is a -closed submodule in , but and . Hence is a -closed submodule in . Proposition 3.5 Let be a non-zero multiplication module, with only one non-zero maximal submodule , then can not be a -closed submodule in . 89 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proof Let be a -closed submodule in , and , , then there exists a -closed submodule in such that and . Thus by [ ], is a weak essential submodule in . But , then by [ ] is a weak essential submodule in . Hence by [ ] , thus , implies that contradiction. Hence dose not be a -closed submodule in . Proposition 3.6 If is a multiplication module over regular ring , and is a non-zero closed submodule in , then is a -closed submodule in . Proof Since is a multiplication module over regular ring , then by [ ] is fully semi-prime. Hence is a -closed submodule in . Proposition 3.7 If is a multiplication module over regular ring , and is a non-zero -closed submodule in , then is a -closed submodule in . Proof Follows by [ ] and proposition ( 2.14 ). Proposition 3.8 If is a multiplication modulr over regular ring , and be a non-zero -closed submodule in and is a -closed submodule in , then is a -closed submodule in . Proof Follows by [ ] and proposition ( 2.16 ). Proposition 3.9 If is a multiplication modulr over regular ring , and are submodules of , with is a -closed submodule in and is a weak essential submodule in , then is a -closed submodule in . Proof Follows by [ ] and proposition ( 2.17 ). Since -closed submodule is -closed , we get the following: Corollary 3.10 If is a multiplication modulr over regular ring , and are submodules of , with is a -closed submodule in and is a weak essential submodule in , then is a -closed submodule in . Proposition 3.11 If is a multiplication modulr over regular ring , and are submodules of , with is a -closed submodule in and is a weak essential submodule in , then is a -closed submodule in . 89 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proof Follows by [ ] and corollary ( 2.20 ). 4. Chain Conditions On -Closed Submodules Module with chain condition on -closed submodules are studied in this section. Definition 4.1 A module is called a module with ascending ( respectively, descending ) chain condition (briefly ACC respectively DCC) on -closed submodules, if every ascending (respectively, descending) chain of -closed submodules of is finite, i.e. such that . Remarks and Examples 4.2 1. Every Noetherian module has ACC on -closed submodules. 2. Every Artirian module has DCC on -closed submodules. Proposition 4.3 If is a module satisfies ACC (DCC) on -closed submodules of , then satisfiesACC ( DCC ) on -closed submodules of . Proof Let ……., be an ascending chain of -closed submodule of for each . But every -closed submodule is -closed, then is a -closed submodule for each …. Since has ACC on -closed submodules, then such that . Thus has ACC on -closed submodules. For DCC in similarly way. Proposition 4.4 If is a fully semi-prime -module satisfies ACC ( DCC ) on -closed submodules of , then satisfies ACC ( DCC ) on closed submodules of . Proof Let …….,be an ascending chain of closed submodule. where is a closed submodule in for each …. But is a fully semi-prime module, then by proposition ( 2.7 ) is a -closed submodule in for each …. But has ACC on -closed submodules. Hence such that . Thus has ACC on closed submodules. Similarly for DCC. Proposition 4.5 If is a module satisfies ACC ( DCC ) on -closed submodules for each …., with each weak essential extension of is completely essential for each …., then satisfies ACC ( DCC ) on closed submodules for each ….. Proof Let ……., be an ascending chain of closed submodule for each …. Then by proposition ( 2.6 ) is a -closed submodule for each …. But has ACC 88 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 on -closed submodules, then such that . Hence has ACC on closed submodules for each …... Similarly for DCC. Proposition 4.6 Let be an -module and be a submodule of such that , where is any -closed submodule in . If satisfies DCC ( ACC ) on -closed submodules of . Then satisfies ACC ( DCC ) on -closed submodules of . Proof Let ……., be a descending chain of -closed submodules in for each …., and for each …. Then by corollary ( 2.13 ) we have is a -closed submodules in for each …. Hence ……., is a descending chain of -closed submodules in . But has ( DCC ) on -closed submodules, so such that . Hence satisfies ( DCC ) on - closed submodules. Similarly for ACC. Proposition 4.7 If is a fully semi-prime -module such that satisfies ACC ( DCC ) on a non-zero -closed submodules of , then satisfies ACC ( DCC ) on non-zero -closed submodules of . Proof Let ……, be an ascending chain of a non-zero -closed submodules of for each ….. Then by proposition ( 2.14 ) is a -closed submodules in for each …. Hence …, be an ascending chain of a -closed submodules in . But has ACC on -closed submodules, then such that . Hence has ACC on -closed submodules. Similarly for DCC. 5. Conclusions In this article we introduce and study the notion of a pseudo weakly closed submodules as a generalization of a -closed submodules. Among the main results we get are the following. 1. If is an -module, and are submodules of with and is a -closed submodule in and is a -closed submodule in , then is a -closed submodule in , provided that contained in any weak-essential extensions of . 2. If is a uniserial -module, and are submodules of such that and is a -closed submodule in and is a -closed submodule in , then is a - closed submodule in . 3. If is an -module where and are submodules of , provided that and all submodules of are completely essential submodules. If is a non-zero -closed submodule in and is a non-zero -closed submodule in , then is a -closed submodule in if and only if is a - closed submodule in and is a -closed submodule in . 88 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 4. Let be a finitely generated, faithful and multiplication - module which satisfies condition (*). Then is a -closed ideal in if and only if is a -closed submodule in . 5. 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