IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 On Max-Modules A. J. Abdul– Al– Kalik Ministry of Education Vocational Education, Khalis , Industrial School Received in: 15, December, 2009 Accepted in: 17, June, 2010 Abstract In this paper ,we introduce a concept of Max– module as follows: M is called a Max- module if Nann R is a maximal ideal of R, for each non– zero submodule N of M; In other words, M is a Max– module iff (0) is a *- submodule, where a proper submodule N of M is called a *- submodule if ][ :KN R is a maximal ideal of R, for each submodule K contains N properly. In this paper, some properties and characterizations of max– modules and *- submodules are given. Also, various basic results a bout Max– modules are considered. Moreover, some relations between max- modules and other types of modules are considered. Key word: Ring, Module, Max-Module Introduction Every ring considered in this paper will be assumed to be commutative with identity and every module is unitary. We introduce the following: An R– module M is called a max- module if Nann R is a maximal ideal of R, for every non-zero submodule No f M, where annR N = {r: r ∈R and r N = 0}. Our concern in this paper is to study max-modules and to look for any relation between max– modules and certain types of well– Known modules specially with primary modules. This paper consists of three sections. Our main concern in §1, is to define and study *- submodules. Also we study the properties of a multiplication module that contains *- submodules. In §2, we study max– modules, and we give some characterizations for this concept. Also other basic results about this concept are given. In §3, we study the relation between max– modules and primary modules. It is clear that every max-module is primary module, but the converse is not true in general. We give in (3.2), (3.3) conditions under which the two concepts are equivalent. Next we investigate the relationships between max, prime, semi– primary, quasi-primary finitely generated and uniform modules, see (3.4), (3.12). 1. SUBMODULES In this section, we introduce the concept of *- submodule and we give some characterizations for this concept. And we end this section by studying the properties of a multiplication module that contains *- submodules. Definition 1.1: A proper submodule N of an R-module M is said to be a *- submodule if ][ :KN R is a maximal ideal of R for each submodule Ko f M such that K N. Where [NR : K] = {r ∈ R: rK ⊆ N}. Specially, an ideal I is a *- ideal of R if and only if I is a *- R– submodule of R– module R.   IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Examples and Remarks (1.2) 1- Recall that an R– submodule N of M is a quasi– primary submodule of M if [NR : K] is a primary ideal of R for each submodule K of M such that K N,[2]. It is well– Known that if ][ :KN R is a maximal ideal of R, then [NR :K] is a primary ideal of R, [1, prop. 4.9, P. 64]. Thus every *- R-submodule of M is a quasi– primary submodule . 2- The submodule Z of the Z-module Q is not a *- submodule since ZZZZ Z 66)6/1(:  is not a maximal ideal of Z . 3- The intersection of any two *- submodules of an R– module need not be *-submodule for example. The Z– module Z6 has two *- submodules, )2(1 N and )3(2 N , but )0(21 NN  is not a *- submodule of Z6, since ZZZZ 66])0[( 6 :  is not a maximal ideal of Z . 4- Every *- submodule is a semi-primary submodule. Proof : Suppose N is a *- submodule of an R-module M. Hence ][ : KN R is a maximal ideal of R. Therefore ][ :KN R is a prime ideal of R, which implies that N is a semi– primary submodule of M by [ 2, definition 1.1 ]. However the converse is not true in general as the following example shows : Let M = Z  Z12 as a Z– module and N=(0)=(0)  (0). It is clear that N is a semi–primary submodule of M, since 00])0[( : MZ is a prime ideal of Z. But (0)  (0) is not a *- submodule of M, since ZZZZ 612])0()0()0[( 12 :  which is not a maximal ideal of Z. By using (1.2, (1)) and [2, Th. (3.1.3), chapter 3] we can give the following characterization for *- submodule. Theorem 1.3 Let N be a proper submodule of an R-module M. If N is a *- submodule of M, then ][ :KN R = ][ :rKN R for each submodule K of M such that K N, rK N and r  R . By using (1.2, (1)) and [2, prop.(3.1.4), chapter 3] we can give the following result : Corollary 1.4 Let N be a proper submodule of an R- module M . If N is a *- submodule of M, then )]([)]([ :: mNrmN RR  for each m ∈ M\N, r ∈ R and r ∉ [NR :(m)]. The converse of corollary (1.4) is not true in general for example: Let M = Z as a Z– module, let N = 6Z, r =5, 5 ∉ [6ZZ :(1)] = 6Z and  ZZZ 6)]1.5(6[ : )]1(6[6 : ZZZ  . But N is not a *- submodule of z. Recall that an R-module M is called a multiplication module, if for every submodule N of M, there exists an ideal I of R such that IM = N, equivalenty ; for every submodule N of M, N= [NR : M] M, see [3] . An R- submodule N of M is called a prime R– submodule if and only if N≠M and whenever r  N , for r  R and x  M, either r  [NR :M] or x  N, [10]. The prime radical P(N) of N in M is defined to be the intersection of all prime submodules P of M such that N ⊆ P i.e. P(N) = ⋂ {P ⊆ M: Pis prime and N ⊆ P}. It is Known that if M is multiplication module and N is a submodule of M, then ][)( :MNNP R M, [ 3, Th. 2. 12 ]. The following remark shows that a multiplication R-module which has a finitely generated *- submodule is finitely generated R– module. Remark 1.5 Let M be a multiplication R-module. If M contains a finitely generated *- submodule N, then M is a finitely generated R– module.      – IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Proof : Since N is a *- submodule, so N is a semi- primary submodule of M by (1.2, (3)). Therefore, M is finitely generated by [2, proposition 3.4, P. 135] . Corollary 1.6 If N is a *- submodule of a multiplication R– module M, then rad (N) is a prime submodule of M. Proof :Suppose that N is a *- submodule. Hence, N is a quasi– primary submodule by (1.2, (1)). But M is a multiplication R– module, so N is a primary submodule of M by [2, propostion (3.1.5), chapter 3]. Therefore, rad (N) is a prime submodule by [4, corollary 2.13, chapter 2] . 2. Basic Properties of Max-Modules In this section, we introduce the concept of a max– module and give some characterizations and properties of this concept, we end the section by studying the relationships between max- rings and max-modules. Definition 2.1 An R– module M is said to be a max– module if NannR is a maximal ideal of R, for each non– zero submodule N of M. Specially, a ring R is called a max– ring if and only if R is max– R– module. We give some examples and remarks: Remarks and Examples 2.2 1- P Z as Z– module is a max– module. N = IM for some ideal I of R. But M is faithful, annRN = annRIM = annRI and so IannIMannNann RRR  which is a maximal ideal of R. Therefore M is a max– module. Proof : We know that every submodule of P Z is of the form  z P n 1 , where n be a non- negative integer, so PZZPZ P ann n nZ  1 is a maximal ideal of Z. 2- Z as a Z– module is not a max– module, since 00 ZannZ is not a maximal ideal of Z. 3- Consider, the Z– module M = Z2  Z12 and the Z– submodule )2()0( N . Then, ZZZZNannZ 666   , which is not a maximal ideal of Z. Therefore, M is not a max– module . 4- Q as a Z– module is not a max– module . 5- Every non– zero submodule of a max– module is a max– R-module. 6- Let M be a max– module, then MannR is a maximal ideal of R. The following theorem gives a characterization for max– modules. Theorem 2.3 Let M be an R-module, then M is a max– module if and only if (0) is a *- submodule. Proof : Suppose that M is a max– module, to prove (0) is a *- submodule. Since M is a max, then NannR is a maximal ideal of R, for each non– zero submodule N of M. But ])0[( : NNann RR  , for each non– zero submodule N of M so by definition (1.1) ,(0) is a *- submodule of M. Conversely, if (0) is a *- submodule of M, to prove M is a max– module. Since (0) is a *- submodule,then definition (1.1) implies that ])0[( : NR is a maximal ideal , for each non-zero submodule N of M . But NannN RR ])0[( : , so M is a max – module. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 By using (1.2, (1)) and [2, theorem (3.3.6), chapter 3], we can give the following characterization for max-module. Theorem 2.4 Let M be an R-module, if M is a max– module then rNannNann RR  for each non– zero submodule N of M such that rN ≠ (0), r  R . By using (1.2, (1)) and [2, corollary (3.3.7), chapter 3], we can give the following result: Corollary 2.5 Let M be an R-module, if M is a max– module then )()( rmannmann RR  for each 0 ≠ m  M such that rm ≠ 0, r  R. Now, we state and prove the following result. Proposition 2.6 Zm as a Z– module is a max– module if and only if m = p n for some prime number p and n  Z+. Proof : If Zm is a max– Z-module, to show that m = pn for some prime number p and n  Z+. By (2.2, [5]), PZmzZmannZ  is a maximal ideal of z, therefore m = pn for some prime number p and n  Z+. Conversely, if m = p n for some p (prime number) and n  Z+, to show that Zm a Z– module is a max– module. Let N be anon– zero submodule of Zm. Since N ⊆ Zm, PZZPmzZmannNann n ZZ  which is a maximal ideal, then PZNannZ  , and by definition (2.1), Zm as a Z- module is a max– module. In the following result, we show that the converse of (2.2. [5]) is true. Proposition 2.7 Let M be an R-module that satisfies ✪,then M is max– module if and only if MannR is a maximal ideal of R. Where ✪: ][ :MNMann RR   , for each non- zero submodule N of M . Proof : If M is a max– module, then by (2.2,[5]) MannR is a maximal ideal of R. Conversely, if MannR is a maximal ideal of R, to prove that M is a max– module, ( NannR is a maximal ideal of R,∀0 ≠ N⊆ M). It is clear that MannNann RR  …..(1). Let Nannr R , so rn N = 0 for some n  Z+ . By ✪, there exists a  R, a ≠ 0 such that aM ≠ 0 and aM  N. Hence rnaM  rnN = 0. It follows that .Mannar R n  But MannR is a maximal ideal, so MannR is a primary ideal by (1, proposition 4.6, P. 64), and a  annRM (since aM ≠ 0), so (r n ) k annRM for some K  Z+ and hence Mannr R . Thus, MannNann RR  …..(2). Therefore, by (1) and (2) we get NannMann RR  . Thus NannR is a maximal ideal and so by definition (2.1), M is a max– module. We note that if M is a max– module, then it is not necessary that R is a max– ring, for example: the Z– module Z2 is max– module, but Z is not max– ring. Moreover , if R is a max– ring and M is an R– module, then M is not necessarily max– module, for example: Consider the Z2– module Z6, Z2 is a max– ring, but Z6, is not max– module. Recall that an R– module M is called faithful R– module if annRM = 0. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 However, in the class of faithful multiplication module, they are equivalent as the following result shows . Proposition 2.8 If M is a faithful multiplication R– module, then M is a max– module if and only if R is a max– ring. Proof : If M is a max– module. To prove R is a max– ring. Let I be a non– zero ideal of R. Then N = IM is a non– zero submodule of M. Hence NannR is a maximal ideal of R because M is a max– module. On the other hand, since M is a faithful multiplication R– module, then annRN = annRI, so IannNann RR  . Thus IannR is a maximal ideal and R is a max– ring. Conversely, if R is a max– ring, to prove M is a max– module. Let N be a non–zero submodule of M. Since M is a multiplication R– module, 3. Some Relations Between Max– Modules And Other Modules In this section, we study the relationships between max-modules and primary modules and prime modules, semi-primary, quasi-primary, finitely generated and uniform modules. We start with the following definitions which are needed. Recall that an R-module M is said to be a primary module if (0) is a primary R– submodule of M, [2] . Where a submodule N of an R– module M is called a primary submodule if N ≠ M and whenever rx  N for r  R and x  M we have either x  N or r n  [NR : M] for some n  Z+, where [NR : M] = {r:r  R ^ rM  N}, [8] . By using this concept, we have the following : Remark 3.1 Every max– module is a primary module. Proof : Let N be a non– zero submodule of an R– module M. Suppose that M is a max– module, to prove M is a primary module. Since M is a max– module, then NannR is a maximal ideal of R, for each non– zero submodule N of M by definition (2.1) and so MannR is a maximal ideal of R by (2.2,6). But MannNann RR  so MannNann RR  . Therefore M is a primary R– module by (2,Theorem (2.1.3), chapter 2). Note that, the converse of (3.1) is not true in general. For example, the Z–module M =Z  Z is a primary by [2, (2.1.2, (2)), Chapter 2], but it is not a max– module. In the following proposition, we give a sufficient condition under which the converse of (3.1) is true. Proposition 3.2 Let M is a module over a PID, and 0 ≠ annRM is a primary ideal of R. If M is a primary R– module, then M is a max– module. Proof : Let N be a non-zero R– submodule of M, to prove NannR is a maximal ideal. Since M is a module over a PID, then the only primary ideals in R are (0) and for some a prime element P and n  z+. But 0 ≠ annRM is a primary ideal, so annRM =

, and this implies  PPMann n R which is a maximal ideal. But M is a primary, then MannNann RR  by [2, Theorem (2.1.3), chapter 2]. Hence NannR is a maximal ideal and so by definition (2.1) M is a max– module. In the following result, we give another condition for which a primary module be a max– module. But first we need the following definition. The dimension of R, denoted by dim R, is defined to be: sup {n  N: there exists a chain of prime ideals of R of length n, if the supremum exists, and ∞, otherwise}, [1]. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Proposition 3.3 Let R be a 0– dimensional ring. Then a primary R– module M is a max– module. Proof : Since M is a primary module, so annRM is a primary ideal of R by (2, corollary 2.1.7, chapter 2) and hence MannP R is a prime ideal. But dim R = 0 implies that p is a maximal ideal. On the other hand, NannMann RR  for every non– zero submodule N of M (since M is a primary module), so that NannR is a maximal ideal. Therefore M is a max– module. Now, we study the relation between max-modules and prime modules. But first we need the following definitions: Recall an R– module M is said to be a prime module if (0) is a prime R–submodule of M, see [9] . We notice that not every max– module is a prime– module, for example: The Z– module Z4 is max by proposition (2.6), but it is not a prime Z-module by [5, (1.1.3 (3)), chapter 1]. The following proposition shows that (annRM is a semi– prime ideal) is a sufficient condition for max– module to be prime. Proposition 3.4 If M is a max-module and annRM is a semi– prime ideal of R, then M is a prime R– module. Proof : Since M is a max– module, then M is a primary R– module by (3.1). But annRM is a semi– prime ideal of R, hence by [2, proposition (2.3.2), chapter 2], M is a prime R– module. Next, a proper submodule N of M is called semi– prime submodule if for every r  R, x  M, K  Z+, such that r k x  N, than rx  N, see [7] . By using this concept, we have the following: Corollary 3.5 If M is a max– module and (0) is a semi– prime submodule , then M is a prime R-module. Proof : Since (0) is a semi-prime submodule, so annRM is a semi– prime ideal by [8, proposition (1-5), chapter 2], hence the result follows by (3.4). Recall an R– module M is said to be a semi– primary if (0) is a semi– primary R– submodule, (2). It is well known that every primary R– module is a semi– primary module [2, (3.5.3, (2)), chapter 3]. So that following result follows immediately from (3.1). Corollary 3.6 Every max-module is a semi– primary R– module. Note that the converse of (3.6) is not true in general. For example, the Z– module M =Z  Z12 is a semi– primary, but not a max– module. Recall that an R-module M is said to be a quasi– primary module if annRN is a primary ideal of R, for each non– zero submodule N of M, [2] . However, we have the following : Remark 3.7 Every max-module is a quasi– primary module. proof : Since M is a max– module, then NannR is a maximal ideal of R for each non–zero submodule N of M . Hence annRN is a primary ideal by [1, proposition 4.9, P. 64], and so M is a quasi– primary. Note that, the converse of (3.7) is not true in general, for example, the Z– module Z is a quasi– primary since annZ(N) = 0 is a prime ideal, for each non– zero N of Z, so it is a primary ideal. But it is not a max– module by [2.2, (2)]. We notice that not every max-module is finitely generated, for example: Z as a Z-module is a max– module but not finitely generated. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 However, we have the following proposition : Proposition 3.8 If M is a multiplication max-module, then M is a finitely generated module. Proof : Since M is a max– module, then MannR is a maximal ideal by [2.2, (6)] and so annRM is a primary ideal by [1, prop. 4.9, P. 64]. On the other hand M is a multiplication imply , M is a finitely generated by [5, prop.(2.7), chapter 2] . Now, we study the relation between max-modules and uniform modules . But first we need the following definition: Recall that an R– module M is said to be uniform module if every non– zero submodule of M is essential, [11] . Where a submodule N of an R– module M is called essential proved that N ∩ K ≠ 0 for every non– zero submodule K of M, [11] . Note that, it is not necessary that every uniform R– module is a max– module for example = Q as a Z– module is uniform. But it is not a max– module by [2.2, (4)] . However, we have the following result. Proposition 3.9 If M is a max– module such that annR(N ∩ U) = annRN + annRU, for every non–zero submodules N and U of M, then uniform. Proof : Since M is a max– module, so M is a primary module by (3.1), hence the result follows by [2, proposition (2.3.7), chapter 2] . Now we can give the following result : Proposition 3.10 Let M be an R-module and let 0 ≠ x  M such that: 1. Rx is an essential submodule of M. 2. )(xannR is a maximal ideal of R, and 3. )(xannMann RR  . Then M is a max – module. Proof : Let N be a non– zero submodule of M. Since Rx is an essential submodule of M, there exists 0 ≠ t  R such that 0 ≠ tx  N and hence (tx)  N. This implies that annRN  annR(tx) and so , )(txannNann RR  . But N  M, then NannMann RR  and hence Nannxann RR )( (by condition 3). Thus, )()( xannNannxann RRR  ……(1). Let )(txannr R , then rntx = 0 for some n  Z+ and rnt  annR(x). But tx ≠ 0; that is t  annR(x) and by condition (2) )(xannR is a maximal ideal of R, so annR(x) is a primary ideal of R, by [1, proposition 4.9, P. 64] . Then )(xannr R and hence )()( xanntxNann RR  ..…(2). Thus by (1) and (2), )()( txannxann RR  and so )(xannNann RR  . Therefore (by condition 2) NannR is a maximal ideal of R and M is a max– module by definition (2.1). The following result is a consequence of proposition (3.10). Corollary 3.11 Let M be uniform R-module such that )(xannR is a maximal ideal of R and )(xannMann RR  for some x ≠ 0. Then M is a max– module. In the following corollary, we give a condition under which the converse of proposition (3.9) is true. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Corollary 3.12 If M is a uniform R– module such that )(xannR is a maximal ideal of R for some x  M. Then the following statements are equivalent. 1. )(xannMann RR  for some x  M. 2. M is a max– module. References 1. Sharp, R.Y. (1990) Steps in commutative algebra London Mathematical Society Student Texts No 19 (Cambridge University Press) . 2. Mijbass, A.S.(2000) "Semi– Primary Submodules", Scince journal, University of Tikrit, J.6. No.1 3. El- Bast, Z.A. and Smith, P.F.(1988) Multiplication modules, Comm. in Algebra, 16: 755- 779 . .معة بغدادجاحول مودیوالت جزئیة في مودیوالت جدائیة، رسالة ماجستیر، )1992( عبد الرحمن عبود احمد .4 5. Abdul- Razak, H.M , Quasi-prime Modules and 'Quasi-prime Submodules' M. D. Thesis, Univ. of Baghdad. 6. Dauns, J. (1980) Prime modules and one– sided ideals in "Ring theory and Algebra III", (Proceedings of Third Oklahoma Conference), B. R. Mc Donald (editor), Dekker, New York, 301-344. .المودیوالت الجزئیة االولیة والمودیوالت الجزئیة شبه االولیة، رسالة ماجستیر، جامعة بغداد )1996( ایمان علي عذاب .7 8. Lu, C.P.(1989) M-radicals of submodules in modules, math. Japon, 34: 211-219. 9. Saymach, S.A. (1979) On prime submodules, University Noc. Tucumare. Ser. A. 29: 121- 136. 10. Ahmed Abdul– Rahman, A. and Al-Hashimi, B. (1994) "On Submodules of Multiplication Modules", Iraqi. J. Sci., 35: 4. 2011) 2( 24المجلد مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة حـول مـقـاس أعـظـم عدویه جاسم عبد الخالق إعدادیة الخالص الصناعیة -التعلیم المهني -وزارة التربیة 2009، كانون االول، 15: استلم البحث في ,2010حزیران ، 17: قبل البحث في الخالصة Maxفي هذا البحث قدمنا مفهوم مقاس مـن النـوع . Rمقاسا أحادیًا على Mحلقة أبدالیة ذات محاید، ولیكن Rلتكن NannNannRadإذا كان ) Max(مقاسًا Mیطلق على : يكما یأت RR )( مثالیًا أعظمیا فيR لكل مقاس جزئي ، وقد أطلقنا على أي مقـاس *مقاسًا من النوع ) 0(إذا كان ) Max(مقاسًا M، بعبارة مكافئة، یكونMفي Nغیر صفري ][إذا كان *مقاسًا من النوع Mفي Nجزئي فعلي :KN R یحتـوي Mفي K، لكل مقاس جزئي Rمثالیا أعظمیا في N ًفي هذا البحث، أعطیت بعض الخواص و التمیزات وكذلك ُدرست العدید من النتـائج األساسـیة حـول المقاسـات مـن . فعلیا والمخطــط اآلتــي یوضـــح . هــذا ُدرســت بعــض العالقــات بینـــه وبــین أنــواع أخــرى مــن المقاســات فضــال عــن ).Max(النــوع : ملخص لما حصلت علیه مودیول، اكبر، المودیل، الحلقة:الكلمات المفتاحیة Max Primary Quasi-Primary Semi-Primary