IBN AL- HAITHAM J. FOR PURE & APPL. S CI. 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 Accepte d in: 17, June , 2010 Abstract In this p aper ,we introduce a concept of M ax– module as follows: M is called a M ax- module if Nann R is a maximal ideal of R, for each non– zero submodule N of M ; In other words, M is a M ax– module iff (0) is a *- submodule, where a prop er submodule N of M is called a *- submodule if ][ : KN R is a maximal ideal of R, for each submodule K contains N p rop erly . In this p aper, some p rop erties and characterizations of max– modules and *- submodules are given. Also, various basic results a bout M ax– modules are considered. M oreover, some relations between max- modules and other ty p es of modules are considered. Key word: Ring, M odule, M ax-M odule Introduction Every ring considered in this p aper 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 p aper is to st udy max-modules and to look for any relation between max– modules and certain ty p es of well– Known modules sp ecially with p rimary modules. This p aper consists of three sections. Our main concern in §1, is to define and st udy *- submodules. Also we st udy the p rop erties of a multiplication module that contains *- submodules. In §2, we st udy max– modules, and we give some characterizations for this concept. Also ot her basic results about this concept are given. In §3, we st udy the relation between max– modules and p rimary modules. It is clear that every max-module is p rimary 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 invest igate the relationship s between max, p rime, semi– p rimary, quasi-primary finitely generated and uniform modules, see (3.4), (3.12). 1. S UBMODULES In this section, we introduce the concept of *- submodule and we give some characterizations for this concept. And we end this section by st udy ing the p rop erties of a multiplication module that contains *- submodules. De fini tion 1.1: A p rop er 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}. Sp ecially , 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. S CI. VOL.24 (2) 2011 Examples and Remarks (1.2) 1- Recall that an R– submodule N of M is a quasi– p rimary submodule of M if [NR : K] is a p rimary 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 p rimary ideal of R, [1, p rop . 4.9, P. 64]. T hus every *- R-submodule of M is a quasi– p rimary 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 examp le. 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 : Sup p ose N is a *- submodule of an R-module M . Hence ][ : KN R is a maximal ideal of R. Therefore ][ : KN R is a p rime ideal of R, which imp lies that N is a semi– p rimary submodule of M by [ 2, definition 1.1 ]. However the converse is not true in general as the following examp le shows : Let M = Z  Z12 as a Z– module and N=(0)=(0)  (0). It is clear that N is a semi–p rimary submodule of M , since 00])0[( : MZ is a p rime 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), chap ter 3] we can give the following characterization for *- submodule. The orem 1.3 Let N be a p rop er 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), chap ter 3] we can give the following result : Corollary 1.4 Let N be a p rop er 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 examp le: 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 multip lication 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 p rime 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 p rime 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, t hen M is a finitely generated R– module.      – IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 Proof : Since N is a *- submodule, so N is a semi- p rimary submodule of M by (1.2, (3)). Therefore, M is finitely generated by [2, p rop osition 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 :Sup p ose that N is a *- submodule. Hence, N is a quasi– p rimary submodule by (1.2, (1)). But M is a multiplication R– module, so N is a p rimary submodule of M by [2, p rop ost ion (3.1.5), chapter 3]. Therefore, rad (N) is a prime submodule by [4, corollary 2.13, chap ter 2] . 2. Basi c Propertie s of Max-Modul es In this section, we introduce the concept of a max– module and give some characterizations and prop erties of this concept, we end the section by st udy ing the relationships between max- rings and max-modules. De fini tion 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 . Sp ecially , a ring R is called a max– ring if and only if R is max– R– module. We give some examp les 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. The orem 2.3 Let M be an R-module, then M is a max– module if and only if (0) is a *- submodule. Proof : Sup p ose that M is a max– module, to p rove (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 p rove M is a max– module. Since (0) is a *- submodule,then definition (1.1) imp lies 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. S CI. VOL.24 (2) 2011 By using (1.2, (1)) and [2, theorem (3.3.6), chap ter 3], we can give the following characterization for max-module. The orem 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), chap ter 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 st ate and prove the following result. Proposi tion 2.6 Zm as a Z– module is a max– module if and only if m = p n for some p rime number p and n  Z+. Proof : If Zm is a max– Z-module, to show that m = p n for some prime number p and n  Z+. By (2.2, [5]), PZmzZmannZ  is a maximal ideal of z, therefore m = p n for some p rime number p and n  Z+. Conversely , if m = p n for some p (p rime number) and n  Z+, to show that Z m a Z– module is a max– module. Let N be anon– zero submodule of Zm. Since N ⊆ Zm, PZZPmzZmannNann nZZ  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 t hat t he converse of (2.2. [5]) is t rue. Proposi tion 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 p rove 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 r n N = 0 for some n  Z+ . By ✪, there exists a  R, a ≠ 0 such that aM ≠ 0 and aM  N. Hence r n aM  r n N = 0. It follows that .Mannar R n  But MannR is a maximal ideal, so MannR is a primary ideal by (1, p rop osition 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 examp le: the Z– module Z2 is max– module, but Z is not max– ring. M oreover , if R is a max– ring and M is an R– module, then M is not necessarily max– module, for examp le: 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- HAIT HAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 However, in the class of faithful multiplication module, they are equivalent as the following result shows . Proposi tion 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 p rove 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 p rove M is a max– module. Let N be a non–zero submodule of M . Since M is a multiplication R– module, 3. S ome Relations Between Max– Modul es And Othe r Modul es In this section, we st udy the relationship s between max-modules and p rimary modules and p rime modules, semi-primary, quasi-primary, finitely generated and uniform modules. We st art with t he following definitions which are needed. Recall that an R-module M is said to be a p rimary module if (0) is a primary R– submodule of M , [2] . Where a submodule N of an R– module M is called a p rimary 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 p rimary module. Proof : Let N be a non– zero submodule of an R– module M . Sup p ose that M is a max– module, to p rove M is a p rimary 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). Not e that, the converse of (3.1) is not true in general. For examp le, 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 p rop osition, we give a sufficient condition under which the converse of (3.1) is true. Proposi tion 3.2 Let M is a module over a PID, and 0 ≠ annRM is a primary ideal of R. If M is a p rimary R– module, then M is a max– module. Proof : Let N be a non-zero R– submodule of M , to p rove NannR is a maximal ideal. Since M is a module over a PID, then the only p rimary ideals in R are (0) and

for some a p rime element P and n  z+. But 0 ≠ annRM is a p rimary ideal, so annRM =

, and this imp lies  PPMann nR which is a maximal ideal. But M is a primary, then MannNann RR  by [2, Theorem (2.1.3), chap ter 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 p rimary 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 p rime ideals of R of length n, if the sup remum exists, and ∞, ot herwise}, [1]. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 Proposi tion 3.3 Let R be a 0– dimensional ring. Then a p rimary R– module M is a max– module. Proof : Since M is a p rimary module, so annRM is a p rimary ideal of R by (2, corollary 2.1.7, chap ter 2) and hence MannP R is a p rime ideal. But dim R = 0 imp lies that p is a maximal ideal. On the other hand, NannMann RR  for every non– zero submodule N of M (since M is a p rimary module), so that NannR is a maximal ideal. Therefore M is a max– module. Now, we st udy 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 p rime module if (0) is a p rime R–submodule of M , see [9] . We notice that not every max– module is a p rime– module, for examp le: The Z– module Z4 is max by p rop osition (2.6), but it is not a p rime Z-module by [5, (1.1.3 (3)), chap ter 1]. The following p rop osition shows that (annRM is a semi– p rime ideal) is a sufficient condition for max– module to be prime. Proposi tion 3.4 If M is a max-module and annRM is a semi– p rime ideal of R, then M is a p rime R– module. Proof : Since M is a max– module, then M is a primary R– module by (3.1). But annRM is a semi– p rime ideal of R, hence by [2, p rop osition (2.3.2), chapter 2], M is a prime R– module. Next, a prop er submodule N of M is called semi– prime submodule if for every r  R, x  M , K  Z+, such that r k x  N, t han 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– p rime submodule , then M is a prime R-module. Proof : Since (0) is a semi-p rime submodule, so annRM is a semi– p rime ideal by [8, p rop osition (1-5), chapter 2], hence the result follows by (3.4). Recall an R– module M is said to be a semi– p rimary if (0) is a semi– p rimary R– submodule, (2). It is well known that every p rimary R– module is a semi– p rimary module [2, (3.5.3, (2)), chap ter 3]. So that following result follows immediately from (3.1). Corollary 3.6 Every max-module is a semi– p rimary R– module. Not e that the converse of (3.6) is not true in general. For examp le, 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– p rimary 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. p roof : 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 p rimary ideal by [1, p rop osition 4.9, P. 64], and so M is a quasi– p rimary. Not e that, the converse of (3.7) is not true in general, for examp le, the Z– module Z is a quasi– primary since annZ(N) = 0 is a p rime 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 examp le: Z as a Z-module is a max– module but not finitely generated. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 However, we have the following p rop osition : Proposi tion 3.8 If M is a multip lication 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 p rimary ideal by [1, p rop . 4.9, P. 64]. On the other hand M is a multiplication imp ly , M is a finitely generated by [5, p rop .(2.7), chap ter 2] . Now, we st udy 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 p roved that N ∩ K ≠ 0 for every non– zero submodule K of M , [11] . Not e that, it is not necessary that every uniform R– module is a max– module for examp le = Q as a Z– module is uniform. But it is not a max– module by [2.2, (4)] . However, we have the following result. Proposi tion 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 p rimary module by (3.1), hence the result follows by [2, p rop osition (2.3.7), chapter 2] . Now we can give the following result : Proposi tion 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 imp lies 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 r n tx = 0 for some n  Z+ and r n t  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, p rop osition 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 p rop osition (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 p rop osition (3.9) is true. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. 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 st atements are equivalent. 1. )( xannMann RR  for some x  M . 2. M is a max– module. Re ferences 1. Sharp, R.Y. (1990) Steps in commutative algebra London M athematical Society Student Texts No 19 (Cambridge University Press) . 2. M ijbass, 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) M ultip lication 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-p rime M odules 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. M c 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. Jap on, 34: 211-219. 9. Saymach, S.A. (1979) On p rime submodules, University Noc. Tucumare. Ser. A. 29: 121- 136. 10. Ahmed Abdul– Rahman, A. and Al-Hashimi, B. (1994) "On Submodules of M ultip lication M odules", 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 Mفي هذا البحث قدمنا مفهوم مقاس مـن النـوع . Rمقاسا أحادیًا على Mحلقة أبدالیة ذات محاید، ولیكن Rلتكن ax M(مقاسًا Mیطلق على : يكما یأت ax ( إذا كانNannNannRad RR )( مثالیًا أعظمیا فيR لكل مقاس جزئي ، M(مقاسًا M، بعبارة مكافئة، یكونMفي Nغیر صفري ax ( إذا كان)وقد أطلقنا على أي مقـاس *مقاسًا من النوع ) 0 ][إذا كان *مقاسًا من النوع Mفي Nجزئي فعلي : KN R مثالیا أعظمیا فيR لكل مقاس جزئي ،K فيM یحتـوي N ًفي هذا البحث، أعطیت بعض الخواص و التمیزات وكذلك ُدرست العدید من النتـائج األساسـیة حـول المقاسـات مـن . فعلیا M(النــوع ax.( والمخطــط اآلتــي یوضـــح . هــذا ُدرســت بعــض العالقــات بینـــه وبــین أنــواع أخــرى مــن المقاســات فضــال عــن : ملخص لما حصلت علیه مودیول، اكبر، المودیل، الحلقة:الكلمات المفتاحیة M ax Primary Quasi-Primary Semi-Primary