Microsoft Word - 262-270 Mathematics | 262 2016) عام 2العدد ( 29مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 End--Prime Submodules Nuhad S. AL-Mothafar Dept. of Mathematics / College of Science/ University of Baghdad. Adwia J. Abdil -Khalik Dept. of Mathematics/ College of Science/ Al-Mustansiriya University. Received in:10/November/2015,Accepted in:6/March/2016 Abstract Let R be a commutative ring with identity and M an unitary R-module. Let (M) be the set of all submodules of M, and : (M)  (M)  {} be a function. We say that a proper submodule P of M is end--prime if for each   EndR(M) and x  M, if (x)  P, then either x  P + (P) or (M)  P + (P). Some of the properties of this concept will be investigated. Some characterizations of end--prime submodules will be given, and we show that under some assumtions prime submodules and end--prime submodules are coincide. Key Words: Prime submodules, S-prime submodules, -prime submodules, end--prime submodules. Mathematics | 263 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 1- Introduction Throughout this paper, R is a commutative ring with identity and M is an unitary R- module. Prime ideals play an essential role in ring theory. One of the natural generalizations of prime ideals which have attracted the interest of several authors in the last two decades is the notion of prime submodule,[1],[2]. These have led to more information on the structure of the R-module. For an ideal I of R and a submodule N of M let√ denote the radical of I, and [N:M] =]{r  R: rM  N} which is clearly an ideal of R. Then a proper submodule P of M is called a prime submodule if r  R and x  M with rx  P implies that r  [P:M] or x  P,[3]. Equivalently P is a prime submodule of M if and only if [P:M] is a prime ideal of R and the R/[P:M]- module M/P is torsion free where the R-module X is said to be torsion free if the annihilator of any nonzero element of X is zero, [3]. There are several generalizations of the notion of prime submodules ,such as Ebrahimi Atani ,F. Farzalipour ,introduced and studied weakly prime submodules, where a proper submodule P of M is said to be weakly prime submodule of M if r  R and x  M, 0  rx  P gives that r  [P:M] or x  P,[4]. A submodule PM is almost prime submodule if r  R and x  M with rx  P\[P:M]P implies that r  [P:M] or x  P,[5]. So any prime submodule is weakly prime and any weakly prime submodule is an almost prime submodule. Another generalization of prime submodule is the concept of S-prime submodules, where a proper submodule p of M is said to be S-prime submodule of M if f(m) P ,where fS=End(M) and mP implies that either mP or f(M)  P,[6].Also in [7] studied S-prime (Endo-prime) submodules. Every S-prime submodule is prime but not conversely ,[6],[7]. Khaksari and Jafari in [8] extended the notion of prime submodule to -prime. Let M be an R-module and (M) be the set of all submodules of M and : (M)  (M)  {} be a function. A proper submodule P of M is said to be -prime if r  R and x  M, rx  P\(P) implies that r  [P:M] or x  P. In this paper ,we define and study the notion of end--prime submodules. Let (M) be the set of all submodules of M and : (M)  (M)  {} be a function. A proper submodule P of M is said to be end-- prime if for each   EndR(M) and x  M, if (x)  P, then either x  P + (P) or (M)  P + (P). 2-Basic Properties of end--Prime Submodules First we give the following definition. Definition (2.1): Let M be an R-module and (M) be the set of all submodules of M. Let :(M)  (M)  {} be a function. A proper submodule N of M is said to be end-- prime if for each   EndR(M) and x  M, if (x)  N, then either x  N + (N) or (M)  N + (N). Remarks and Examples (2.2): (1) It is clear that every S-prime submodule is end--prime submodule .The convers is not true as the following example shows .Let M = Z8 as Z- module ,N ={0 ,4 }.Then N is not S- prime submodule of M ( since if f( ̅) = 2 ̅ , ̅Z8 where f :Z8 Z8 and f(2) = 2.2 = 4N. But 2 N and f(M) ={0,2, 4 ,6 }⊈ N, hence N is not S- prime submodule of M .But N is end  - prime submodule of M . Proof: Let: (Z8) (Z8)  {}, where (N)= N +  2 ,  N M and for all f : Z8 Z8 .If ƒ( ̅ )N={ 0 , 4 },then either ̅ N+(N) =2  or f (Z8)  N + (N) = 2.Therefore N is end--prime submodule of Z8. Mathematics | 264 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 (2) Let M = Z4 as Z-module, N = {0, 2} . Then N is an end--prime submodule of M (since N is S- prime submodule of M,[6]) . (3) The only end--prime submodule of a simple module is {0}.Therefore(0) in the simple Z-module Zp (p is prime number) is end--prime submodule. (4) It is clear that not every end--prime submodule is prime submodule, see example in remark(2.2,1). (5) If (N) = N or (N) = 0,then every end--prime submodule is S-prime submodule and hence is prime submodule. (6) Let M = Z12 as Z-module, then N = {0 , 6 is not end--prime of M .Since if f : Z12  Z12, where f ( ) = 2 for all  Z12 and let :(M)  (M)  {} such that (N) =N + {0 ,6} , N M . Since f (3) = 6  N, but 3  N + (N)= {0 ,6 }and f (Z12 ) = 2Z12 ⊈N + (N) = {0 ,6 }.Therefore N={0 , 6 is not end--prime submodule of Z12. Recall that an R-module M is called scalar if for every f  End(M),  r  R, r  0 such that f (m) = rm for all r  R, [9]. The following proposition shows that( scalar R-module ) is a sufficient condition for prime submodule to b end--prime submodule. Proposition (2.3): Let M be a scalar R-module, and N is a prime submodule of M. Then N is an end-- prime submodule of M. Proof: Let f  End(M), m  M such that f (m)  N. Since M is scalar,  r  R, r  0 such that f (x) = rx for all x  M. Hence f (m) = rm  N. But N is prime, so either mN or rM N. Thus either m  N + (N) or f (M)  N + (N). Therefore N is end--prime submodule . Corollary (2.4): Let N a prime submodule of a finitely generated multiplication R-module M .Then N is an end--prime submodule of M . Proof: By [9,corollary 1.1.11]"Every finitely generated multiplication R-module M is a scalar module " and so by proposition (2.3) we get the result. Corollary (2.5): If N is a prime submodule of a cyclic R-module M, then N is end--prime submodule of M. Proof: By [6,porosition 2.1.4],we have N is an S-prime submodule of M and by remark (2.2,1),we have the result. Recall that a submodule N of an R-module M is said to be fully invariant if f (N)  N, for each R-endomorphism f of M ,[10]. By using this concept ,we can give the following result. Theorem (2.6): Let N be a proper fully invariant submodule of an R-module M. If [N+(N): f (M)]=[N+(N): f (K)], for all N+(N) ⊊K and for all f  End(M) such that f ((N) ) = (N) then N is an end--prime sumodule of M. Proof: Let h(m)  N, where h  End(M) and m  M and suppose that m  N + (N), we must prove that h(M)  N + (N). Now, N +(N) ⊊ N +(N) + Rm hence by assumption [N+(N):h(M)] = [N+(N):h(N +(N)+ Rm)]. But 1  [N+(N):h(N +(N)+ Rm)] since h(N) +h((N)) + h(Rm)  N +(N), therefore 1  [N+(N):h(M)], which implies that h(M)  N + (N). Therefore N is an end--prime of M. Mathematics | 265 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 Recall that "An R-module M is called duo if for each submodule N of M ,N is fully invariant ,[11]. The following result follows immediately from theorem (2.6). Corollary (2.7): Let M be an duo R-module .If [N++(N): f (M)]=[N+(N): f (K)], for all N+(N)⊊ K and for all f  End(M) such that f ((N) ) = (N) , then N is an end--prime submodule M. Proposition (2.8): If N is an end--prime submodule of R-module M and (N) < N ,then [N +(N) : f (M)] = [N+(N) : f (k)], for all N+(N) ⊊   K and for all f  End(M). Proof: Since N is an end--prime of M and (N) < N, so by remark (2.2,5) N is S-prime .Hence by [6,prop.(2.1.14)] ,[N: f (M)] = [N: f (k)], for all f  End(M) and N ⊊ K .Since (N) < N ,then [N +(N) : f (M)] = [N+(N) : f (k)], for all and N+(N) ⊊⊊   K. and for all f  End(M). Recall that a submodule N of an R –module M is called relatively divisible (S- relatively divisible ) denoted by RD(S-RD) if r M  N = r N for each r  R , ƒ(M)  N=ƒ(N) for all f  End(M), [6],[10] respectively . Recall that a nonzero module M is called quasi-dedekind if Hom (M/N,M) = 0 for all nonzero submodule of M. Equivalently, M is quasi-dedekind if for any f  End(M), f  0, then ker f = {0} (i.e. f is 1-1), [12]. Proposition (2.9): Let M be a quasi –Dedekind R-module .Then every proper S-RD submodule of M is end--prime submodule of R-module M . Proof: By [13,prop1.12],every proper S-RD submodule of M is strongly S-prime and by [14,rem.(1.2,2)],every strongly S-prime submodule is S-prime submodule .This implies every proper submodule of an R-module M is end -prime submodule by [Rem.(2.1,1)] . More About end--Prime Submodules In this section, several fundamental properties of end--prime submodule are given. Proposition (3.1): Let M be an R-module, N < M, I  R. If P is an end--prime submodule of M such that IN P , then N  P + (P), provided I ⊈ [P + (P):M]. Proof: Suppose IN P and I ⊈ [P + (P):M], Let x  N, we must prove that x  P + (P) for any x  N . Since I ⊈ [P + (P):M], then there exists a  I and a  [P + (P):M]. Define f : M  M by f (m) = a m for all m  M, it is clear that f  EndR(M) and f (x) = a x  IN  P. But P is an end--prime submodule of M and f (M) = aM ⊈ P + (P), so x  P + (P). Thus N  P + (P). Proposition (3.2): Let M be an R-module, let   End(M). If N is fully invariant end--prime of an R- module M, such that  (M) ⊈ N and ( – 1(N)) =  – 1 ((N)) , then  – 1(N) is also end-- prime submodule of M. In fact in this case  – 1(N) = N. Mathematics | 266 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 Proof: First, we must show that  – 1(N) is a proper submodule of M. Suppose that  – 1(N) = M, then (M)  N, which a contradiction to the assumption. Now, let f (m)   – 1(N), where f  End(M) and m  M. If m   – 1(N) + ( – 1(N)), then (m)  N +  (( – 1(N))), which implies that m  N + (N), since N is fully invariant submodule of M and  – 1 =  – 1. We only have to show that f (M)   – 1(N) + ( – 1(N)). Since f (m)   – 1(N), then (◦f )(m) =  ( f (m))  N. But N is an end--prime of M and m  N + (N), therefore (◦f )(M)  N + (N). This implies f (M)   – 1(N) +  – 1((N)) =  – 1(N) + ( – 1(N)). Recall that an R-module M is called A-projective (where A is an R-module) if for every X < A and every homoorphism : M  A X can be lifted to a homomorphism : M  M, [14]. If M is A-projective for each R-module A, then M is called projective. Theorem (3.3): Let f: M  M' be an epimorphism and let N < M such that ker f  N. If N is an end-- prime submodule of a module M, then f (N) is an end-'-prime submodule of a module of M', where M' is M-projective module and '( f (N)) = f ( (N)). Proof: First, we must show that f (N) is a proper submodule of a module M'. Suppose f (N) = M'. But f is an epimorphism, thus f (N) = f (M) and hence M = N + ker f. This implies that M = N. A contradiction. Now, let h(m')  f (N), where h  End(M') and m'  M' and suppose that m'  f (N) + '( f (N)), we have to show that h(M')  f (N) + '( f (N)). Since f is an epimorphism and m'  M', then there exists m  M, such that f (m) = m'  f (N) + '( f (N)), thus m  N + f – 1('( f (N))) = N + (N). Consider the following diagram: M' k h M M' 0 f since M' is M-projective module, then there exists a homoorphism k: M'  M, such that f ◦k = h. Clearly, k◦ f  End(M). Note that f (k◦ f (m)) = (f ◦k)( f (m)) = h(m')  f (N) and since ker f  N, we get (k◦ f )(m)  N. But N is an end--prime submopdule of M and m  N + (N). Therefore (k◦ f )(M)  N + (N) and hence k( f (M)) = k(M')  N + (N). Thus f (k(M'))  f (N) + f ((N)), which implies that h(M')  f (N) + '( f (N)). Corollary (3.4): Let M be an R-module, let K < N < M and N is an end--prime. Then N K is end-'- prime in M K , provided that M K is M-projective. Mathematics | 267 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 Recall that an R-module M is A-injective (where A is an R-module) if for every X  M, any homomorphism : X  M can be extended to a homomorphism : A  X, [9], [15]. If M is M-injective M is called quasi-injective, [9]. Proposition (3.5): Let K be an end--prime of an R-module M and let N < M which is M-injective and (K)  K. Then either N  K or K  N is an end--prime in N. Proof: Suppose that N ⊈ K, then K  N is a proper submodule in N. Let f (x)  K  N, where f  End(N) and x  N. Suppose x  (K  N) + '(K  N), where ': (N)  (N){} be a function, then x  K. We must show that f (N)  (K  N) + '(K  N). Now, consider the following diagram: i 0 N M f h N Where i is the inclusion map. Since N is M-injective, then there exists h: M  N, such that h◦i = f, clearly h  End(M). But f (X) = (h◦i)(x) = h(x)  K. Since K is an end--prime submodule of M and x  K + (K), this implies that h(M)  K + (K). Also, f (N) = (h◦i)(N) = h(N)  N (since f (N)  K  N) and f (N)  h(N)  h(M)  K +(K). Therefore, f (N)  N  (K + (K)) = N  K  N  K + '(N  K). Corollary (3.6): Let K be an end--prime submodule of a quasi-injective R-module M, and let N < M. Then either N  K or K  N is an end--prime in N. Proposition (3.7): Let M be an R-module and let K < N < M and K is fully invariant. If N K is an end-'- prime submodule of M K and N N K K        , then N is an end--prime submodule of M. Proof: Suppose that f (m)  N, where f  End(M) and m  M. If m  N + (N), then we must show that f (M)  N + (N). Define f *: M K  M K by f *(x + K) = f (x) + K,  xM. To prove f * is well define, let x + K = y + K where x, y  M, then x – y  K and hence f (x – y)  f (K)  K, since K is fully invariant. This implies that f (x) – f (y)  K. Thus f (x) + K = f (y) + K. Now, f *(m + K) = f (m) + K  N K . But N K is an end--prime of M K and m + K  N N K K          hence M N N * K K K f              and thus (M) K K f   N N K K        and which implies that (M) K K f   N N N K K K       , thus f (M) + K  N + (N) and f (M)  N + (N). . Mathematics | 268 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 Proposition (3.8): Let M be a projective R-module. If N is end--prime and (N) = 0  N < M, then M N is a quasi-Dedekind R-module. Proof: To prove M N is quasi-dedekind, we shall prove any endomorphism on M N is either zero mapping or 1-1, let f : M N  M N and f  0. Since M is projective there exists h: M  M such that ◦h = f ◦, where  is the natural projection. Hence for any m  M, (◦h)(m) = (h(m)) = h(m) + N = ( f ◦)(m) = f (m + N). If f (x + N) = M N 0 = N for some x + N  M N , then h(x) + N = N and so h(m)N. Hence either x  N + (N) or h(M)  N , since N is end--prime. Thus, either x + N = N = M N 0 ,or [(h(M))=0 ,then( f ∘)(M) = f (M/N) =0 which is a contradiction] .Therefore f is 1-1- and M N is quasi-dedekind. For a partial answer for the converse of Prop.(3.8) we have the following: Proposition (3.9): Let N < M such that N is fully invariant such that M N is a quasi-Dedekind R-module. Then N is end--prime. Proof: Let f  End(M) and f (m)  N for some m  M. Define g: M N  M N by g(x + N) = f (x) + N,  x  M, g is well-defined. If g = 0, then f (M)  N  N + (N). If g  0, then g is 1-1 and hence g(m + N) = f (m) + N = N implies that m + N = N; that is m  N  N + (N). Thus N is end--prime. Mathematics | 269 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 References 1. Dauns, J., (1978), Prime Modules, J. Reine . Angew .Math., 298, 156-181. 2. Moore, M.E. and Smith, S.J., (2002), Prime and Radical Submodules of Modules Over Commutative Rings, Comm. Algebra, 30, 5037-5064. 3 . Lu, C.P., (1981),Prime Submodules of Modules, Commutative Mathematics ,University Spatula,33,16-69. 4. Atani ,S.E .and Farzalipour, F.,(2007),On Weakly Prime Submodules, Tamkang Journal of Mathematics ,38(3),247-252. 5. Khshan,H.A.,(2012),On Almost Prime Submodules , Acta Mathematics Scientia ,32B(2),645-651. 6. Shireen ,O.D.,(2010),S-prime Submodules and Some Related Concepts, M.Sc. Thesis ,College of science ,University of Baghdad. 7. Gungoroglu ,C., (2000),Strongly Prime Ideals in CS-Rings ,Turk .J.Math., 24,233-238. 8. Khaksari, A. and Jafari, A., (2011), -Prime Submodules, International Journal of Algebra, 9. Shihab, B.N., (2004), Scalar Reflexive Modules, Ph.D. Thesis, Univ. of Baghdad. 10. Faith, C.,(1973), Rings, Modules and Categories, I, Springer , Berlin, Iteidelberg , New York . 11. Abbas ,M.S.,(1990),On Fully Stable Modules, Ph. D. Thesis, University of Baghdad. 12. Mijbass, A.S.,( 1997), Quasi-Dedekind Modules and Quasi-Invertible Submodules, Ph.D. Thesis, Univ. of Baghdad. 13. Inaam ,M.A.,(2011),Strong S-Prime Submodules , Almustansiriyah J .of Scines.22,201- 210. 14. Azumaya, G., Mbuntum, F. and Varadarajan, K.J., (1975), On Projective and M-Injective Modules, 95, 9-16. 15. Mohamed, S.H. and Muller, B.J., (1990), Continuous and Discrete Modules, Cambridge University Press Cambridge. Mathematics | 270 2016) عام 2العدد ( 29لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 29 (2) 2016 End-المقاسات الجزئية األولية من النمط نهاد سالم المظفر جامعة بغداد /كلية العلوم /قسم الرياضيات قعبد الخالعدويه جاسم الجامعة المستنصرية /كلية العلوم /قسم الرياضيات 2016اذار//6قبل في: ،2015تشرين الثاني//10استلم في: خالصةال مجموعة كل المقاسات (M). لتكن Rمقاساً معرفا ً على الحلقة Mحلقة ابدالية ذات عنصر محايد، وليكن Rلتكن هو مقاس Mمن Pدالة. في هذا البحث، نقول ان المقاس الجزئي : (M)  (M)  {}ولتكن Mالجزئية من ، فانه يؤدي الى (x)  Pا ذ ان End(M) ،x  Mاذا كان لكل End-جزئي أولي من النمط . لقد درسنا واعطينا بعض خواص و مميزات هذا النوع من المقاسات (M)  P + (P)أو x  P + (P)اما 0ونان متكافئين الجزئية وبرهنا تحت شروط معينة ان المقاسات الجزئية االولية وهذا النوع من المقاسات الجزئية يك ، المقاسات الجزئية االولية من S -المقاسات الجزئية االولية ، المقاسات الجزئية االولية من النمط :الكلمات المفتاحية .End-، المقاسات الجزئية االولية من النمط النمط