2011) 2( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد المقاسات الجزئیة األولیة المضادة هادي انعام محمد علي جامعة بغداد،ابن الهیثم -كلیة التربیة،قسم الریاضیات 2010 ،شباط ،28: استلم البحث في 2010 ،نیسان، 25: قبل البحث في الخالصة یقـال عـن . Mمقاس جزئـي فعلـي مـن Nلیكن . Rمقاساً احادیاً على Mحلقة ابدالیة ذو محاید ولیكن Rلتكن N مقاساً جزئیاً اولي مضاد اذا كان المقاس   اولي مضاد، حیث ان المقاس   rیسمى اولي مضاد اذا كـان لكـل  R اما ،O     r أو     r. .في هذا البحث درسنا المقاسات الجزئیة األولیة المضادة واعطینا العدید من الخواص المتعلقة بهذا المفهوم المقاســات -)المضـادة االولیـة(قاســات الجزئیـة الثانیـة المقاسـات الثانیــة الم -المقاسـات الجزئیــة االولیـة المضـادة :الكلمـات المفتاحیـة .الثانویة IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 Coprime Submodules I. M. A. Hadi Departme nt of Mathematics, Ibn-Al-Haitham, College of Education , Unive rsity of Baghdad Received in: 28, February, 2010 Accepte d in: 25, April, 2010 Abstract Let R be a commutative ring with unity and let M be a unitary R-module. Let N be a p rop er submodule of M , N is called a coprime submodule if   is a coprime R-module, where   is a cop rime R-module if for any r  R, either O     r or      r . In this p aper we st udy coprime submodules and give many p rop erties related with this concept. Key words: Cop rime submodules, second submodule, second (coprime) module, secondary module. Introduction Let R be a commutative ring with unity and let M be a unitary R-module. It is well- known t hat a p rop er submodule N of an R-module M is called p rime if whenever rR, xM , rxN implies xN or r  [N:M ], where [N:M ]={rR: rMN}. M is called a p rime module if R ann M = R ann N for all nonzero submodule N of M , equivalently M is a p rime module iff (0) is a prime submodule. S.Yassem in [7], introduced the notions of second submodules and second modules, where a submodule N of M is called second if for any r R, the homothety r*End M , is either zero or surjective, where r*(m) = r m,  m  M . It follows that N is a second submodule iff for each r R, either rN = 0 or rN = N. M is called a second module if M is a second submodule of itself. For an R-module M , the following st atements are equivalent: (1) M is a second module. (2) For each r R, either rM = 0 or rM = M . (3) ann M = ann   for all prop er submodules N of M . (4) ann M = ann   for all fully invariant sub3 (5) modules N of M . (6) ann M = W(M ), where W(M )={r R:r*End M , r* is not surjective}. Not ice (1)  (2) is clear, (1)  (5) [7,lemma 1.2], (1)  (3) [3, theorem 2.1.6], (3)  (4) [6, theorem 1.3.2]. Not ice that st atement (3) and st atement (4) are used to define coprime module by S. Annin in [2] and I.E Wijayart in [6], resp ectively. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 M oreover Rasha in [3] st udied coprime modules and give some generalizations of these modules, (see [3]). J.Abuhilail in [1], introduced the notion of coprime submodule, where a p rop er submodule N of M is called coprime if ann   = W(   ); that is N is a coprime submodule if   is a cop rime R-module. Our aim in this p aper is t o st udy coprime submodules, we give the basic prop erties about this concept. Also, we study coprime submodules in certain classes of modules. 1- Coprime S ubmodul es We give the basic p rop erties related with coprime submodules. Also, we st udy their behaviour in certain classes of modules. Following J.Abuhilail in [1], a p rop er submodule N of an R-module M is called coprime if   is a cop rime R-module. An ideal I of a ring R is called coprime ideal iff R  is a coprime R-module. 1.1 Remarks and Examples: (1) N is cop rime submodule iff for each r  R either O     r = N or      r , that is N is a cop rime submodule if for each r R, either r  [N:M ] or for any m  M , there exists m'  M such that m – r m'  N. (2) Z is a coprime submodule of the Z-module Q, since Q Z is a coprime Z-module [4], [6]. Not e that Z is not coprime Z-module, since when r = 2  0, 2Z  Z. (3) Every submodule N of the Z-module p is a coprime submodule, since p /N  p and p is a cop rime Z-module, hence p /N is a coprime Z-module. (4) Let M be a coprime R-module, then every p rop er submodule N of M is a coprime submodule. proof: Since M is a coprime R-module, then by [3,cor. 2.1.12],   is a coprime R- module, for all N < M . Hence N is a coprime submodule. (5) If N is a maximal submodule of an R-module M , then N is a cop rime submodule. proof: Since N is maximal,   is a simple R-module, hence   is a coprime R- module. Thus N is a coprime submodule. (6) The converse of (4) is not true in general for examp le, Z is a coprime submodule of the Z- module Q (see 1.1 (2)) but Z is not a maximal submodule of Q. (7) Let M be an R-module, let I be an ideal of R such that I  ann M , let N < M . Then N is a coprime R-submodule of M  N is a coprime R -submodule of M , where R =R / I.   IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 proof: () Let N be a cop rime R-submodule. Then   is a coprime R-module and hence by [3, cor. 2.1.9],   is coprime R -module. Thus N is a cop rime R -module. () The proof is similarly. 1.2 Proposi tion: If N is a coprime submodule, then [N:M ] is a prime ideal. proof: Since N is a coprime submodule,   is coprime R-module. Hence ann   is a p rime ideal of R [3, note 2.1]. But ann   = [N:M ], so [N:M ] is a prime ideal. Recall that an R-module M is called secondary if for each r  R, either r m = 0 or r n M = M , for some n  Z+. [7]. We have the following: 1.3 Proposi tion: Let M be a secondary R-module, let N < M . Then N is a cop rime submodule iff [N:M ] is a prime ideal of R. proof: () It follows by p rop . 1.2. () Since M is a secondary R-module, then   is a secondary R-module. But [N:M ] = ann   is a p rime ideal, so by [3,p rop .1.2.6],   is a coprime R-module, hence N is a coprime submodule. 1.4 Proposi tion: Let N be a p rop er submodule of an R-module M . Then N is a coprime submodule iff [N:M ]=[W:M ] for all W  N. proof: If N is a coprime submodule, then   is a coprime R-module. Hence ann   = ann W    for all WN. It follows that ann   =ann W  ; that is [N:M ]= [W:M ]. If [N:M ] = [W:M ], for all W  N, then ann   =ann W  . But W   W    , so ann   =ann W    and   is a cop rime R-module. Thus N is a cop rime submodule. 1.5 Proposi tion: Let W be a coprime submodule of M and let N < M such that N  W. Then N is a coprime submodule of M and W  is a cop rime submodule of M W .   IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 proof: Since W is a coprime submodule, then M W is a coprime R-module. Hence by [Rem and Ex. 1.1 (4)], W  is a coprime submodule of M W . Also M W is a coprime R-module imp lies (M /W) / (N/W) is a cop rime R-module [3,cor. 2.1.12]. But (M /W) / (N/W)  M / N, hence M / N is a coprime module by [3, Cor. 2.1.14]. Thus N is a cop rime submodule of M . 1.6 Proposi tion: Let M be an R-module, let N, W be p rop er submodules of M , N  W such that W  is a coprime submodule of M W .Then N is a cop rime submodule of M . proof: Since W  is a coprime submodule of M W , we have (M /W) / (N/W) is a coprime module. Thus M / N is a cop rime module and so N is a coprime submodule of M . The following results follow directly by p rop osition 1.5. 1.7 Corollary: If N is a coprime submodule of an R-module M , I an ideal of R. Then [N :  I] is a coprime submodule of M . 1.8 Corollary: Let A, B be p rop er submodules of an R-module M . If A or B is a coprime submodule and A + B  M . Then A + B is a coprime submodule of M . 1.9 Proposi tion: Let I be a p rop er ideal of a ring R. Then I is a coprime ideal iff I is a maximal ideal of R. proof: If I is a coprime ideal of R, then R/I is a cop rime R-module. But R/I is a multiplication R-module, so by [3,Rem. And Ex. 2.1.3(5)] R/I is simple R-module. Thus I is a maximal ideal of R. The converse follows by (Rem. And Ex. 1.1.(5)). 1.10 Corollary: Let R be a ring. The following are equivalent: (1) (0) is a cop rime submodule of R. (2) R/(0) � R is a cop rime ring (that is R is a field). (3) (0) is a maximal ideal of R. 1.11 Corollary: Let R be a PID, let I be a nonzero prop er ideal of R. Then the following are equivalent: (1) I is a cop rime ideal of R. (2) I is a maximal ideal of R. (3) I is a prime ideal of R. 1.12 Note : If N is a coprime submodule of an R-module M . Then it is not necessary that [N:M ] is a coprime ideal of R, as the following examp le shows: Z is a cop rime submodule of the Z-module Q but [Z:Q] = (0) is not a maximal ideal of Z, that is (0) is not coprime ideal of Z. 1.13 Proposi tion: Let M be a multiplication R-module, let N be a p rop er submodule of M . Then N is a coprime submodule iff [N:M ] is a cop rime ideal of R. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 proof: If N is a coprime submodule of M , then   is a coprime R-module. But M is a multiplication R-module imp lies   is a multiplication R-module. Hence by [3,Rem. and Ex. 2.1.3(5)]   is a simple R-module. Thus N is a maximal submodule of M which imp lies that [N:M ] is a maximal ideal. Then by p rop . 1.9, [N;M ] is a cop rime ideal. Conversely , if [N:M ] is a cop rime ideal of R, then by p rop . 1.9, [N:M ] is a maximal ideal of R. Now M is a multiplication module and [N;M ] is a maximal ideal of R imp lies that N=[N;M ]M is a maximal submodule of M . Thus by Rem. and Ex. 1.1 (5), N is a coprime submodule of M . 1.14 Corollary: Let M be a multiplication R-module and let N < M . The following are equivalent: (1) N is a coprime submodule of M . (2) [N:M ] is a cop rime ideal of R. (3) [N:M ] is a maximal ideal of R. (4) N is a maximal submodule of M . proof: (1)  (2) it follows by p rop . 1.13. (2)  (3) it follows by p rop . 1.9. (4)  (1) by Rem. and Ex. 1.1 (5). (3)  (4) Since M is multiplication, and [N:M ] is a maximal ideal, then N is a maximal submodule of M . The following result shows that a homomorp hic image of a coprime submodule is a coprime submodule. 1.15 The orem: Let :M  be an R-ep imorp hism, let N < M . If N is a coprime submodule of M , then (N) is a coprime submodule of  . proof: To p rove (N) is a cop rime submodule of  , we must p rove ( )    is a cop rime R- module, so we must show that ( ) ( )         r for all ann ( )     r . First ann ( )     r , means that [ ( ) : ]   r . It is easy to check that [N:M ]  [ ( ) : ]   . Hence [ : ] ann       r . On the other hand N is a coprime submodule, imp lies   is a coprime R-module. Hence      r since ann [ : ]       r . Now, let y + (N)  ( )    , so y = (m) for some m  N, since  is an ep imorp hism. Thus y + (N) = (m) + (N) = ( m + N). Hence there exists m'  M such that. m + N = r m +N, so y + (N) = ( r m' +N) = r (m') + N = r ((m') + N)    r . Thus ( ) ( )         r and so ( )    is a coprime R- module. Hence (N) is a coprime submodule of  . Now, we turn our att ention to direct sum of cop rime submodules. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 1.16 The orem: Let M 1, M 2 be R-modules, let N1 < M 1, N2 < M 2 such that 1 2 ann ann 1 2      . Then N = N1N2 is a coprime submodule of M iff N1 is a coprime submodule of M 1, N2 is a coprime submodule of M 2. proof: () Let p 1:M 1M 2  M 1, p 2:M 1M 2  M 2 be the natural p rojection. Hence p 1(N1N2) = N1, p 2(N1N2) = N2 and so by theorem 1.15, N1 is a cop rime submodule of M 1, N2 is a cop rime submodule of M 2. Conversely , to p rove N1N2 is a coprime submodule of M 1M 2. Since N1, N2 are coprime submodules of M 1, M 2 resp ectively, then 1 1   and 2 2   are coprime R-module and since 1 2ann ann 1 2      it follows that 1 2 1 2      is a coprime R-module (see [7], [3,p rop . 2.3.3). But it is easy to check that 1 2 1 2 1 2 1 2           � . Hence by [3,cor. 2.1.14], 1 2 1 2       is a coprime R-module. Thus N1N2 is a coprime submodule of M 1M 2. 1.17 Remark: The condition 1 2ann ann 1 2      is necessary condition in Th. 14, as the following examp le shows: Consider the Z-module Z. Let N1 = 2Z, N2 = 3Z, N1, N2 are maximal submodules of Z, so N1, N2 are coprime submodules of Z (see Rem. 1.1(5)). Let N = N1  N2 = 2Z  3Z < ZZ. It is clear that ann ann 1 2      . Now 2 3 6 1 2 1 2                � � . But Z6 is not a cop rime Z-module, so 1 2      is not a cop rime Z-module. Thus N1  N2 is not a cop rime submodule of ZZ. The following p rop erty exp lains the behaviour of coprime submodules under localization. 1.18 Proposi tion: Let S be a multiplicative subset of a ring R. Let N be a p rop er submodule of an R- module M such that S – 1 N  S – 1 M . If N is a coprime submodule of M , then S – 1 N is coprime sbmodule of S – 1 M . proof: N is a coprime submodule of M imp lies   is a coprime R-module, then by [3,p rop .2.1.38], S – 1         is a cop rime S – 1 R-module. But [5,lemma 9.12,p .173], IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 2) 2011 1 1 1 S S S            , so 1 1 S S     is a coprime S – 1 R-module. Hence S – 1 N is a coprime submodule of S – 1 M . Recall that an R-module M is antihop fian if M = M /N for all N