Microsoft Word - 147-154 147 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 Purely Goldie Extending Modules Saad A. Al-Saadi Ikbal A. Omer Dep. of Mathematics /College of Science/University of Al Mustansiriyah Received in: 4 March 2015, Accepted in : 13 April 2015 Abstract An -module is extending if every submodule of is essential in a direct summand of . Following Clark, an -module is purely extending if every submodule of is essential in a pure submodule of . It is clear purely extending is generalization of extending modules. Following Birkenmeier and Tercan, an -module is Goldie extending if, for each submodule of , there is a direct summand D of such that . In this paper, we introduce and study class of modules which are proper generalization of both the purely extending modules and -extending modules. We call an -module is purely Goldie extending if, for each , there is a pure submodule P of such that . Many characterizations and properties of purely Goldie extending modules are given. Also, we discuss when a direct sum of purely Goldie extending modules is purely Goldie extending and moreover we give a sufficient condition to make this property of purely Goldie extending modules is valid. Key words: extending module, purely extending module, -extending module, purely Goldie extending. 148 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 Introduction Throughout all rings are associative and R denotes a ring with identity and all modules are unitary R-modules. A submodule of a module is called essential if every non-zero submodule of intersects nontrivially (notionally, e M). Also, a submodule of is closed in , if it has no proper essential extension in [1]. Recall that a module M is extending if every submodule of is essential in a direct summand of . Equivalently, every closed submodule of M is direct summand [1]. Many generalizations of extending modules are extensively studied. Following Fuchs [2] and Clark [3], an -module is purely extending if every submodule of is essentialin a pure submodule of M (recall that a submodule N of an R-module M is pure if IM∩N =IN for every finitely generated ideal I of R). Also in [4], the following relations on the set of submodules of an R-module M are considered. (1) if and only if there exists a submodule of such that e A and e A; (ii) if and only if ∩ e X and ∩ e Y. Following [4], is reflexive and symmetric, but it may not be transitive. Also, is an equivalence relation. Moreover, an R-module is extending if and only if for each submodule of , there exists a direct summand of such that [4]. In 2009 Birkenmeier and Tercan [4], an - module is called Goldie extending (shortly, -extending) if, for each submodule of , there is a direct summand D of such that . In section one, we introduce purely -extending modules. An R-module M is -extending if, for each , there is a pure submodule P of such that . It is clear that every - extending (purely extending) module is purely -extending module and the converse is not true in general. Additional conditions are given to make the converse true. In fact we prove that: let be a pure split. Then is a purely -extending module if and only if is a – extending module. Moreover, the hereditary property of purely -extending modules is discussed. We call an R-module M is purely -extending if every direct summand of M is purely – extending. We do not know whether every purely –extending module is purely - extending. Indeed, we conclude that every purely extending module is purely -extending. Finally, we prove that an Z-module is extending if and only if M is a purely extending and is a -extending. In section two, various characterizations of purely -extending modules are given. For example, we prove that an -module is purely -extending if and only if every direct summand of the injective hull of , there exists a pure submodule of such that ∩ ) . On other direction, the direct sum property of purely –extending modules is discussed. We prove that, if is purely -extending module for each ∈ and every closed submodule of =⊕ ∈ is fully invariant, then =⊕ ∈ is purely -extending module. 1. Purely Goldie Extending Modules. Recall that an -module is -extending if, for each submodule of , there is a direct summand D of such that . Equivalently , is Goldie extending if and only if for each closed submodule C of , there is a direct summand D of such that [4], Also, an - module is purely extending module if every submodule of is essential in a pure submodule of [3]. We introduce and study the class of modules which is a generalization of both -extending modules and purely extending modules. 149 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 Definition (1.1) An -module is called purely Goldie extending (shortly, purely -extending) if, for each , there is a pure submodule P of such that . Remarks and Examples (1.2) 1) Every purely extending module is a purely -extending, but the converse is not true in general. For example, the Z-module ⊕ is a purely –extending since is -extending [4]. But by [4, Example (3.20)] and proposition (1.14), ⊕ is not purely extending Z-module. 2) Every - extending module is purely -extending, but the converse is not true in general. For example, by [5, Example (3.4)], the Z-module ⊕ ∈ Z is purely extending but it is not extending. So M is a purely –extending while, by proposition (1.14), M is not -extending . 3) Every uniform module is purely -extending, but the converse is not true in general. For example, as Z-module is purely - extending but it is not uniform. Recall that an -module is a pure-split if every pure submodule of is a direct summand [6].The following proposition gives conditions under which the concepts of - extending modules and purely - extending modules are equivalent. Proposition (1.3): Let is a pure split -module. Then is a purely -extending if and only if is a - extending. ∎ Following [7], a non-zero -module is pure-simple if the only pure submodules of are 0 and itself. Proposition (1.4) Let be a pure- simple -module. Then is a purely - extending if and only if is a uniform module. Proof:(⟹) Let be a submodule of . By assumption, there is a pure submodule P of such that β . So, ∩ is essential in .But is a pure- simple then = , then is essential in . Thus, is a uniform module. (⟸) Let be a submodule of . Since is a uniform module, then is essential in , but is a pure submodule of , then β . Hence, is a purely - extending. ∎ Corollary (1.5) Let be a pure- simple -module. Then the following statements are equivalent. (1) is a purely extending module. (2) is a purely -extending module. (3) is uniform module. Following [4], a submodule of -extending module need not to be -extending. Moreover, a submodule of purely extending module need not to be purely extending [5]. In fact, we do not know whether a submodule of a purely -extending module is purely -extending. Indeed, we have the following result. Proposition (1.6) Every submodule of a purely -extending -module with the property that the intersection of with any pure submodule of is a pure submodule of is purely - extending. 150 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 Proof : Let be a submodule of . Since is a purely -extending, then there is a pure submodule of such that . By assumption, ∩ is a pure submoduleof .But, ∩ ≤e P and ∩ ≤e A, so ∩ ∩ ≤e ∩ and ∩ ∩ ≤e ∩ =A. Therefore, ∩ . Thus, is purely -extending module. ∎ From [4], recall that is -extending module if every direct summand of is - extending.This lead us to introduce the following. Definition (1.7): An -module is called purely -extending if every direct summand of is purely - extending. In fact, we do not know whether, every purely -extending module is purely - extending. In fact, we have the following result. Proposition (1.8): Every purely extending module is purely -extending module. Proof : Let be a direct summand of a purely extending module . By [5], is purely extending module. Hence is purely -extending module. Thus, is a purely - extending. ∎ But the converse of proposition (1.8) is not true in general, for example, the Z- module ⊕ (for any prime number ) is not purely extending by (1.2), but is purely -extending, since the only direct summands of , ( ⊕ 0 , (0 ⊕ ) , (0 ⊕ 0 and , which are purely -extending. Recall that an R-module M has the pure intersection property (PIP) if the intersection of any two pure submodule of M is pure [8]. Proposition (1.9) : Let be a purely -extending and has the . Then is a purely -extending. Proof : Let be a direct summand of and be a submodule of . Since is a purely - extending, then there is a pure submodule of such that . But satisfies , then ∩ is a pure submodule of . But ∩ ⊆ N, hence ∩ is a pure submodule of .Therefore, ∩ ∩ by [9], and so is a purely -extending. ∎ Corollary (1.10) : Let be a prime module over a Bezout domain. If is a purely -extending module, then is a purely -extending. ∎ Recall that an -module is a multiplication if for each submodule of , there exists an ideal of such that = [10]. Since every multiplication module has the [8]. Thus, we have the next corollary. Corollary (1.11): Let be a multiplication purely -extending module. Then is a purely - extending. ∎ Corollary (1.12) : Let is cyclic module over a commutative ring . If is a purely -extending, then is purely -extending. ∎ Corollary (1.13) : Let be a purely -extending commutative ring, then is a purely - extending. ∎ 151 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 The following result gives a characterization of extending abelian groups. Proposition (1.14): A -module is extending module if and only if is a purely extending and is a - extending as -module. Proof : (⟹) it is clear that . (⟸) Let be a closed submodule of . Since is a purely extending, then is a pure submodule of by [5]. Also, since is a -extending as -module by [4], then is a direct summand of .Therefore, is extending module. ∎ 2. Characterizations of Purely Goldie Extending Modules It is known that is a purely extending module if and only if every closed submodule in is a pure in [5]. Also from [4], is -extending module if and only if for every closed submodule of M, there is a direct summand of such that . Here, we give analogous characterization of purely -extending modules. Proposition (2.1): An -module is purely -extending if and only if for every closed submodule of , there is a pure submodule of such that . Proof : (⟹) it is clear . (⟸) Let be a submodule of .By Zorn's lemma, there exists a closed submodule of such that is essential in . So, we have A .By assumption, there exists a pure submodule of such that . Since is transitive relation, then A .Therefore, is purely -extending module. ∎ Proposition (2.2): An -module is purely -extending if and only if every direct summand of the injective hull , there exists a pure submodule of such that ∩ ) . Proof : (⟹) Let be a direct summand of the injective hull of , then ( ∩ is a submodule of , since is purely -extending,then there exists a pure submodule of such that ∩ . (⟸) Let is a submodule of and let be a relative complement of such that ⨁ is essential in [11]. Since is essential in , then ⨁ is essential in . Thus, ⨁ ⨁ )= [10]. By hypothesis, there exists a pure submodule of such that ( ∩ ) . But is essential in . Therefore, ∩ ≤e ∩ . But ∩ = ∩ ∩ ∩ ≤e ∩ and ∩ = ∩ ∩ ∩ ≤e ∩ . So, ∩ ) ∩ . Since is transitive, then ∩ . So is purely -extending. ∎ Proposition (2.3): The following statements are equivalent for an an - module : (1) is purely – extending module. (2) For each is a submodule of , there exists a submodule of and a pure submodule of , such that and . Proof: (1)⟹(2) Let be a submodule of . Then there exists a pure submodule of such that , so ∩ and ∩ . The proof is complete put ∩ . (2)⟹(1) Let be a submodule of . By (2), there exists a submodule of and a pure submodule of such that and . Now, since ∩ and ∩ then ∩ and ∩ . So and so is purely – extending module. ∎ Following [4], a direct sum of -extending modules need not be -extending module. Also, a direct sum of purely extending modules need not be purely extending module [5]. Here, we discuss when a direct sum of purely -extending modules is a purely -extending. 152 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 Recall that a submodule N of an R -module M is fully invariant if f ( N )  N for each R-endomorphism f of M [12]. M is called Duo if every submodule of M is fully invariant [13]. Proposition (2.4) Let is purely -extending R-module for each ∈ such that every closed submodule of =⊕ ∈ is fully invariant, then =⊕ ∈ is purely -extending module. Proof : Let be a closed submodule of and let : ⟶ be the natural projection on for each ∈ . Let ∈ , so ∑ ∈ , where ∈ and hence ( )= . Now, since is closed submodule of, then by hypothesis , is fully invariant and hence ( )⊆ ∩ . So ( )= ∈ ∩ and hence ∈⊕ ∈ ∩ . Thus ⊆⊕ ∈ ∩ ). Also, ⊕ ∈ ∩ )⊆ and so ⊕ ∈ ∩ ) . Since ∩ )⊆ and by purely - extending property of , then there is a pure submodule of such that ∩ , ∀ ∈ . Now, since is a pure submodule of ,∀ ∈ , then ⊕ ∈ is a pure submodule in ⊕ ∈ [8].So, ⊕ ∈ ∩ ⊕ ∈ [9].Thus, is purely -extending module. ∎ Corollary (2.5) : Let ⨁ be a duo module such that and are purely -extending modules. Then is a purely -extending. ∎ By the same argument of the proof proposition (2.4), one can get the following result. Firstly, recall that an R- module M is distributive if for all submodules K, L and N of M, K ∩ (L + N) = (K ∩ L)+ (K ∩ N)[14]. Proposition (2.6) Let ⨁ be a distributive module such that and are purely -extending modules. Then is a purely -extending. Proof: Let is a submodule of ⨁ since is a distributive module so A A⋂M A⋂ ⨁ A⋂ ⨁ A⋂ . But and are purely - extending, then there are a pure submodule of such that A⋂ and pure submodule of such that A⋂ . So, A A⋂ ⨁ A⋂ ⨁ by [9] and by [8] ⨁ is a pure submodule of ⨁ . Thus, is a purely -extending. ∎ Proposition (2.7): Let and be purely -extending -modules such that . Then ⊕ is a purely -extending module. Proof : Let (≠0) be a submodule of ⊕ . Since , then ⊕ , where is a submodule of and is a submodule of [15]. Since (≠0) then (≠0) or (≠0).If ≠0 and =0, then = is a submodule of . But M is purely -extending and hence there is a pure submodule of such that . Since is a direct summand of ⊕ , then is a pure submodule of ⊕ , (by [16]), then pure submodule of ⊕ .Thus ⊕ is a purely -extending module. By the similar way if =0 and ≠0, then ⊕ is a purely -extending module. If (≠0) and (≠0), since and are purely - extending modules, then there is a pure submodule of such that , and there is a pure submodule of such that . But ⊕ is a pure submodule of ⊕ [8] and by [9], ( ⊕ ⊕ . Therefore, ⊕ is a purely -extending module. ∎ 153 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 References 1. Dung, N.V.; Huynh, D.V.; Smith, P.F. and Wisbauer R.: (1994), Extending modules, John Wiley and Sons, Inc. New York. 2. Fuchs, L.: (1995), Notes on generalized continuous modules, preprint. 3. Clark, J.: (1999), on purely extending modules, In Abelian groups an modules. Proceedings of the international conference in Dublin, Ireland, August 10-14, 1998 (ed. By Eklof, Paul C, etal.), Basel, Birkhasure, Trends in Mathematics, 353-358. 4. Akalan, E., Birkenmeier G. F. and Tercan A. , (2009), Goldie Extending modules, Comm. Algebra 37: 2, 663-683. 5. Al- Zubaidey , Z. T.: (2005), On purely extending modules, MSc. Thesis, Univ. of Baghdad . 6. Azumaya, G. and Faccini A.: (1989), Rings of pure global dimension zero and Mittag- leffler modules, J.pure Appl. Algbra, 62, 109-122. 7. Fieldhouse, D.J.: (1969), pure theories, Math. Ann. 184, 1-18. 8. Al- Bahraany B.H.: (2000), modules with pure intersection property, Ph.D. Thesis, Univ. of Baghdad. 9. Enas, M. Kamil: (2014), Goldie Extending (Lifting) modules, MSc. Thesis, Univ. of Baghdad. 10. Barnad, A.: (1981), Multiplication modules, J. Algebra 71, 174- 178. 11. Anderson, F.W. and Fuller K.R.: Rings and Categories of modules, Springer-Verlag. New York 1973. 12. Wisbauer, R.: (1991), Foundations of Module and Ring theory, reading: Gordon and Breach Science Publishers. 13. Lam, T.Y.: (1988), Lectures on Modules and rings, Springer-Verlag, Berin, Heidelberg. New York. 14. Erdogdu, V. : (1987), Distributive Modules, Can. Math. Bull 30, 248-254. 15. Abbas, M. S.: (1991), On fully stable modules, Ph.D. Thesis, Univ. of Baghdad. 16.Yaseen, S.H. : (1993), F-Regular Modules, M.Sc. Thesis, University of Baghdad. 154 | Mathematics ٢٠١٥) عام ٢العدد ( ٢٨المجلد والتطبيقية الھيثم للعلوم الصرفة ابنمجلة Ibn Al-Haitham. J. for Pure & Appl. Sci. Vol. 28 (٢) 2015 -مقاسات التوسع النقية من النمط سعد عبد الكاظم الساعدي قبال احمد عمرإ المستنصرية العلوم / الجامعةقسم الرياضيات/ كلية ٢٠١٥نيسان ١٣في: البحث قبل ،٢٠١٥أذار ٤ :في البحث استلم الخالصة يكون Mبأنه توسع إذا كان كل مقاس جزئي من Mيقال للمقاس R.مقاسا ً معرفا ًعلى Mحلقة و Rلتكن يكون Mبأنه توسع نقي إذا كان كل مقاس جزئي من M. تبعاً كالرك، يقال للمقاس Mجوھريا ً من مركبة جمع مباشرمن . -. من جھة اخرى، بركانمير و تيركان عرضا مفھوم مقاسات التوسع من النمطMجوھريا ً من مقاس جزئي نقي من بحيث Mمن Dيوجد مركبة جمع مباشر Mمن Xأذا كان لكل مقاس جزئي - بأنه توسع من النمط Mيقال للمقاس . في ھذا البحث، تم عرض و دراسة صنف من المقاسات كتعميم فعلي لكل من صنف مقاسات التوسع النقية Mمن Xإذا كان لكل مقاس جزئي -بأنه توسع نقي من النمط M. نقول عن المقاس -ومقاسات التوسع من النمط . تم أعطاء العديد من التشخيصات و النتائج و الخواص لمقاسات التوسع بحيث Mن م Pيوجد مقاس جزئي نقي مقاس توسع –. وكذلك تم مناقشة متى تكون مركبة الجمع المباشرلمقاسات التوسع النقية من النمط -النقية من النمط . –صية متحققة لمقاسات التوسع النقية من النمط . أكثر من ذلك، تم تقديم شروط كافية لجعل ھذه الخا - نقي من النمط ، مقاسات التوسع النقية من النمط - مقاسات التوسع من النمط النقية،مقاسات التوسع التوسع،مقاسات :المفتاحيةالكلمات – .