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 ـات الجزئیـة الثانیـة المقاسـات الثانیــة الم -المقاسـات الجزئیــة االولیـة المضـادة :الكلمـات المفتاحیـة المقاســات -)المضـادة االولیـة(قاسـ .الثانویة IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Coprime Submodules I. M. A. Hadi Department of Mathematics, Ibn-Al-Haitham, College of Education , University of Baghdad Received in: 28, February, 2010 Accepted 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 proper submodule of M, N is called a coprime submodule if   is a coprime R-module, where   is a coprime R-module if for any r  R, either O     r or      r . In this paper we study coprime submodules and give many properties related with this concept. Key words: Coprime 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 that a proper submodule N of an R-module M is called prime 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 prime module if R ann M = R ann N for all nonzero submodule N of M , equivalently M is a prime 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 statements 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 proper 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}. Notice (1)  (2) is clear, (1)  (5) [7,lemma 1.2], (1)  (3) [3, theorem 2.1.6], (3)  (4) [6, theorem 1.3.2]. Notice that statement (3) and statement (4) are used to define coprime module by S. Annin in [2] and I.E Wijayart in [6], respectively. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Moreover Rasha in [3] studied coprime modules and give some generalizations of these modules, (see [3]). J.Abuhilail in [1], introduced the notion of coprime submodule, where a proper submodule N of M is called coprime if ann   = W(   ); that is N is a coprime submodule if   is a coprime R-module. Our aim in this paper is to study coprime submodules, we give the basic properties about this concept. Also, we study coprime submodules in certain classes of modules. 1- Coprime Submodules We give the basic properties related with coprime submodules. Also, we study their behaviour in certain classes of modules. Following J.Abuhilail in [1], a proper submodule N of an R-module M is called coprime if   is a coprime 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 coprime submodule iff for each r  R either O     r = N or      r , that is N is a coprime 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]. Note 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 coprime Z-module, hence  p /N is a coprime Z-module. (4) Let M be a coprime R-module, then every proper 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 coprime 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 example, 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. SCI. VOL.24 (2) 2011 proof: () Let N be a coprime R-submodule. Then   is a coprime R-module and hence by [3, cor. 2.1.9],   is coprime R -module. Thus N is a coprime R -module. () The proof is similarly. 1.2 Proposition: 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 prime 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 rnM = M , for some n  Z+. [7]. We have the following: 1.3 Proposition: Let M be a secondary R-module, let N < M . Then N is a coprime submodule iff [N:M] is a prime ideal of R. proof: () It follows by prop. 1.2. () Since M is a secondary R-module, then   is a secondary R-module. But [N:M] = ann   is a prime ideal, so by [3,prop.1.2.6],   is a coprime R-module, hence N is a coprime submodule. 1.4 Proposition: Let N be a proper 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 coprime R-module. Thus N is a coprime submodule. 1.5 Proposition: 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 coprime submodule of M W .   IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 implies (M/W) / (N/W) is a coprime 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 coprime submodule of M. 1.6 Proposition: Let M be an R-module, let N, W be proper submodules of M , N  W such that W  is a coprime submodule of M W .Then N is a coprime 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 coprime module and so N is a coprime submodule of M. The following results follow directly by proposition 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 proper 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 Proposition: Let I be a proper 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 coprime 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 coprime submodule of R. (2) R/(0) � R is a coprime 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 proper ideal of R. Then the following are equivalent: (1) I is a coprime 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 example shows: Z is a coprime 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 Proposition: Let M be a multiplication R-module, let N be a proper submodule of M. Then N is a coprime submodule iff [N:M] is a coprime ideal of R. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 implies   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 implies that [N:M] is a maximal ideal. Then by prop. 1.9, [N;M] is a coprime ideal. Conversely, if [N:M] is a coprime ideal of R, then by prop. 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 implies 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 coprime 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 prop. 1.13. (2)  (3) it follows by prop. 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 homomorphic image of a coprime submodule is a coprime submodule. 1.15 Theorem: Let :M  be an R-epimorphism, let N < M. If N is a coprime submodule of M, then (N) is a coprime submodule of  . proof: To prove (N) is a coprime submodule of  , we must prove ( )    is a coprime 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, implies   is a coprime R-module. Hence      r since ann [ : ]       r . Now, let y + (N)  ( )    , so y = (m) for some m  N, since  is an epimorphism. 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 attention to direct sum of coprime submodules. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 1.16 Theorem: Let M 1, M 2 be R-modules, let N1 < M 1, N2 < M 2 such that 1 2ann 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 p1:M1M 2  M 1, p2:M1M 2  M 2 be the natural projection. Hence p1(N1N2) = N1, p2(N1N2) = N2 and so by theorem 1.15, N1 is a coprime submodule of M 1, N2 is a coprime submodule of M2. Conversely, to prove N1N2 is a coprime submodule of M 1M 2. Since N1, N2 are coprime submodules of M 1, M 2 respectively, 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,prop. 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 example 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 coprime Z-module, so 1 2     is not a coprime Z-module. Thus N1  N2 is not a coprime submodule of ZZ. The following property explains the behaviour of coprime submodules under localization. 1.18 Proposition: Let S be a multiplicative subset of a ring R. Let N be a proper submodule of an R- module M such that S – 1 N  S – 1M. If N is a coprime submodule of M, then S – 1N is coprime sbmodule of S – 1M. proof: N is a coprime submodule of M implies   is a coprime R-module, then by [3,prop.2.1.38], S – 1         is a coprime S – 1 R-module. But [5,lemma 9.12,p.173], IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 1 1 1 S S S            , so 1 1 S S     is a coprime S– 1R-module. Hence S– 1N is a coprime submodule of S – 1 M. Recall that an R-module M is antihopfian if M = M /N for all N