- Primary Submodules

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1 - Primary Submodules Nuhad Salim Al-Mothafar*, Ali Talal Husain Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq Abstract Let is a commutative ring with identity and be a unitary -module, a submodule of an -module is called primary if whenever, then either or [ ] for some. In this paper we say that is -primary submodule of an -module if whenever ( ) ( ), then either or ( ( )) [ ] for some, where [ ]. We also study a -primary module if (0) is a -primary submodule of. We give many properties of -primary submodule of and -primary module. Keywords: prime submodule, primary submodule, primary module ISSN: المقاسات الولية - من النمط الخالصة عل ى لتكن نهاد سالم المظفر*, علي طالل حسين قسم الرياضيات, كلية العلوم, جامعة بغداد, بغداد, الع ارق حلقة ابدالية ذات عنصر محايد, وليكن. يقال ان المقاس الجزئي فانه اما او ]. [ مقاس ابتدائي من النمط مقاسا جزئيا من المقاس صفري االيسر اذا كان في هذا البحث نقول ان المقاس الجزئي بحيث ان المعرف ابتدائي من النمط اذا كان ) ( بحيث ) ( فانه اما او (( ( (.[ ]. سندرس ايضا المقاسات االولية من النمط اذا كان ) ( مقاس جزئي ابتدائي من النمط لقد اعطينا العديد من النتائج على هذا النوع من القاسات والمقاسات جزئية 1. Introduction Let be a commutative ring with identity and let be a unitary (left) R-module. A proper ideal in a ring R is called prime ideal if whenever implies that either or [ ].[1]. A proper submodule of an -module is called prime submodule whenever,,, implies that either [ ] or.[2]. A proper ideal of a commutative ring with identity is called primary ideal if for each and, then there exists such that [1]. A proper submodule of an R-module is called primary submodule if whenever such that, then either or [ ], for some [3]. Primary submodule which is generalization to primary ideal is studied by many mathematicians [4]. A proper submodule of an - module is called Z- prime submodule whenever ( ), ( ), implies that either ( ) [ ] or.[5]. In this paper we study the concept of Z-primary submodule as a generalization of the concept of primary submodule, where a proper submodule of an R-module is called Z-primary if whenever = ( ),, such that ( ) then either ( ( ) ) [ ] for some or, also we define Z-primary module if (0) is Z-primary submodule of, and we give some remark and examples of Z-primary submodule and Z- primary module. * nuhadmath@yahoo.com 163

2 2. Some properties of Z-primary submodules Recall that a proper submodule of an -module is called Z-prime, if ( ), ( ),, implies that either ( ) [ ] or.[5] First we give the following definition Definition (2.1): Let be an -module a proper submodule of is called -primary if whenever ( ) for ( ) and we have either ( ( )) [ ] or, for some Remarks and examples (2.2): 1. Every primary submodule is a Z-primary submodule. But the converse is not true in general, for example: let, but or [ ] Thus is not primary, to show is a -primary submodule of, since ( ), then, then 0. Hence is a -primary submodule of. 2. It is clear that every Z-prime is Z-primary. But the converse is not true let as a submodule of as -module, is Z-primary submodule of, since is a primary submodule of, but is not Z-prime, since if we take by ( ) for each, ( ), but 2 and 2 [ ] 3. Consider the Z-module Z. The submodule is a Z-primary submodule if and only if P is a prime number and n a positive integer. since is primary submodule hence is a Z-primary submodule. it is clear since satisfies in primary submodule 4. Consider the Z-module. The submodule ( ) is not a Z-primary submodule, since if we take :, define by ( ) ( ) ( ) ( ) ( ), but [ ] ( ) for each and (3,0) In the following proposition we give when does a -primary submodule to be primary Proposition (2.3): Let be cyclic, faithful R-module and be a submodule of, if is Z-primary submodule of then is primary submodule of. Let where, suppose we want prove that [ ] suppose, then ; t Define : by ( ) ( ), is well define since faithful, this implies that ( ), but is -primary submodule of and, then there exist =( ( )) [ ]. Thus is primary submodule of Corollary (2.4): Let be a PID, if is -primary submodule of cyclic, faithful -module, then [ ] is a - primary ideal of. Let is Z- primary submodule of an -module,since R is cyclic faithful ring then is primary submodule, thus [ ] is primary ideal.[6]. Since is cyclic faithful ring, therefore by proposition (2.3) [ ] is a Z-primary ideal of The converse of corollary (2.5) is not true in general the example show that: Let as Z-module and consider the submodule ( ) of the then [ ] [ ( ) ] which is -primary ideal of Z, since is primary ideal. but not a Z-primary submodule of, since if we take :, define by ( ), ( )( ) ( ) ( ) but [ ( ) ] ( ) for each,and (2,0). Proposition (2.5): Let be an -module and be a submodule of if [ ] [ ] for each submodule of such that, then is a -primary submodule of. To prove is a Z-primary submodule of, let ( ) such that ( ) and suppose, let ( ), then and so ( ) [ ] [ ],thus ( ) [ ] and hence is a Z-primary submodule of 164

3 Proposition (2.6): Let be a proper submodule of an -module, if [ ( )] is a -primary ideal, for each then [ ( )] is a -primary ideal, for all, where ( ) Suppose [ ( )] is a -primary ideal,. To prove [ ( )] is a -primary ideal, for all. Let = ( ),, such that ( ) [ ( )], and suppose that ( ) [ ( )], then ( ) ( ). Hence ( ) then [ ( )] by assumption is a Z-primary ideal of, since ( ) [ ( )] thus ( ) [ ( )], but [ ( )] is a -primary ideal of so either ( ) [ ( )] or [ ( )] for some Thus either ( )( ( ) or ( ( )), however ( )( ( )) implies ( ) ( ) which contradicts our assumption. Thus ( ( )) hence [ ( )] and so [ ( )] and hence [ ( )] is a -primary ideal of for each Proposition (2.7): Let be a proper submodule of an -module, if is a -primary submodule of, then [ ] is a -primary submodule of where is a ideal of R Let be a -primary submodule of such that ( ) [ ] ; ( ) and any ideal of, then ( ) for all, so either for all or ( ( )) [ ] for some. The first case implies that [ ]. The second case implies that ( ( )) but [ ], hence ( ( )) [ ]. It follows that ( ( )) [[ ] ]. Therefore, either [ ] or ( ( )) [ [ )] ] and hence [ ] is a -primary submodule of Recall that an -module is said to be a primary R-module if (0) is a primary submodule of. [7] Definition (2.8): An -module is said to be a Z-primary module if ( ) is a -primary submodule of Proposition (2.9): Let be a nonzero -module if ( ) ( ) for all nonzero submodule of, then ( ) is a -primary submodule of Let ( ) ; ( ),, and suppose is nonzero submodule of an -module and hence by assumption Ann ( ) ( ), but ( ) ( ), thus ( ) ( ). Definition (2.10): [8] An element is called a zero-divisors on an -module, if for some non zero The set of all zero-divisors of on is denoted by Zdvr( ), Proposition (2.11): Let be a nonzero -module if ( ) ( ) then is a -primary module Assume that ( ), suppose. This implies that ( ) ( ) ( ), hence ( ) ( ) Remark (2.12): A direct summand of Z-primary R-module is also Z-primary R-module. Since every non-zero submodule of Z-primary R-module is a Z-primary module Recall that a submodule of an R-module is called pure if. In case is principal ideal domain (PID) or is cyclic, then is pure if and only if, for each,[9] Proposition (2.13): Let R be (PID) and let be a proper submodule of Z-primary R-module. If is a pure submodule of, then is Z-primary submodule of Let, such that ( ) and suppose. Thus ( ) ( ), but is pure submodule of, implies that ( ) ( ) which mean that ( ) ( ) for some 165

4 . Hence ( )( ) and. Since M is a Z-primary R-module, therefore ( ( )) for some. which mean that ( ( )) [ ]. But [ ] [ ], hence( ( )) [ ] for some Therefore is Z-primary submodule of Example (2.14): Let as a Z-module and, where is a prime number, so is a Z-primary submodule of and it is clear that is a Z-primary Z-module. But is not a pure submodule, since if ( ) then (P). But ( ) ( ) 3. More about Z-primary submodules Proposition (3.1): Let be an epimorphism. If is a -primary submodule of, such that, then ( ) is a -primary submodule of Suppose that ( ). But is epimorphism, thus ( ) ( ) and hence one can show easily, but this implies that, which is a contradiction. Now, let ( ) ( ) ( ) ( ), and suppose ( ), we must prove that ( ) [ ( ) ], since is an epimorphism and, then there exist, such that ( ), and ( ) ( ) ( ) = ( ( ) ) ( ) and since, we get ( ), but is a -primary submodule of and, then ( ) [ ] and clearly, [ ] [ ( ) ( )] = [ ( ) ], therefore ( ) [ ( ) ] Corollary (3.2): Let be a Z-primary submodule of, if is any submodule of contained in, then is a - primary submodule of. Proposition (3.3): Let be a Z-primary submodule of an -module and let be a summand submodule of, then either or is Z-prime submodule of Suppose, then is proper in, let ( ), ( ) and, suppose and thus, since is summand there exist a projection and ( ) ( ), since K is Z-primary submodule there exists ( ( )) [ ] [ ] Thus ( ( )), and also( ( )), hence ( ( )), thus ( ( )) [ ]. If is a Z-primary submodule, then is called primary submodule, where [ ]. And hence if (0) is a Z-primary submodule of then (0) is [ ] -Z-primary Proposition (3.4): Let be any ring, let be a Z-primary ideal of, let be a positive integer and let be a P-Zprimary submodule of for each.then is also a P-Z-primary submodule of It is clear that [ ]. Let and such that ( ) and suppose, then there exists an integer with such that. But ( ) and is a P-Zprimary, it follows that ( ). Hence is a P-Z-primary submodule. Remark (3.5): 1. The direct sum of any two Z-primary submodules of an -module need not to be Z-primary submodule of, for example:,. 4Z is a primary submodule of as -module, hence is a Z-primary and (0) is primary, hence is Z-primary submodule of as Z-module but ( ) is not Z- primary in see in Corollary (2.4). 2. It is not necessary that every proper submodule of -primary module is -primary submodule.,. is Z-primary module but is not Z-primary submodule of see in (2.2)(4) 166

5 Proposition (3.6): Let and be two -modules and, if is a -primary submodule of, then and are -primary submodules of and respectively. To prove is a Z-primary submodule of, let ( ),, such that ( ), then ( )( )( ), where, but is a Z- primary submodule of, so either ( ) or ( ( )) [ ] for some, Thus either or ( ( )) [ ] [ ] for some, hence either or ( ( )) [ ] for some, therefore is a Z-primary submodule of, by similar proof is a Z-primary submodule of References: 1. Burton, D.M A first course in rings and ideals, Addison-Wesley Publishing, Company. 2. LU.C.P Prime Submodule of Modules. Comment.Math.Univ.Stpaul, 33, pp: LU.C.P M-radicals of submodule in modules. Math japon, 34, pp: LU, C.P The Zariski Toplogy on the prime specturum of module, Houston J.math., 25(3), pp: Al-Mothafar, N. S Z-prime submodules. M.Sc. Thesis, College of Science, Baghdad University, Iraq. 6. Ali, M.I Primary modules. Ibn al-haitham College, Baghdad University, Iraq. 7. Smith, P.F Primary modules over commutative rings, Glasgow math, 27, pp: Naderi, M.N Weak primary Submodules of Multiplication Modules and Intersection Theorem. Int.J.Contemp.Math.Sciences,4(33), pp: Filedhouse, J.D Pure Theories, Math. Ann., 184, pp: