مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Strongly Essentially Quasi-Dedekind Modules I. M.A.Hadi, T.Y.Ghawi Department of Mathematics, College of Education Ibn-Al-Haitham , University of Baghdad Department of Mathematics, College of Education, University of Al- Qadisiya Received in: 5 April 2011, Accepted in: 13July 2011 Abstract Let R be a commutative ring with unity . In this paper we introduce and study the concept of strongly essentially quasi-Dedekind module as a generalization of essentially quasi- Dedekind module. A unitary R-module M is called a strongly essentially quasi-Dedekind module if 0),( MNMHom for all semiessential submodules N of M. Where a submodule N of an R-module M is called semiessential if , 0 pN for all nonzero prime submodules P of M . Key Words: Essentially quasi-Dedekind Modules; Strongly essentially quasi-Dedekind Modules, Semiessential submodules, Multiplication Modules. 1. Introduction Let R be a commutative ring with unity and M be an R-module. Mijbass A.S in [7] introduced and studied the concept of quasi-Dedekind, where an R-module M is called quasi- Dedekind if, 0),( MNMHom for all nonzero submodules N of M. Ghawi Th.Y. in [4] introduced and studied the concept of essentially quasi-Dedekind, where an R-module M is called essentially quasi-Dedekind if, 0),( MNMHom for all essential submodules N of M (N e M ). In this paper we give a generalization of essentially quasi-Dedekind which we call it strongly essentially quasi-Dedekind, where an R-module M is called strongly essentially quasi-Dedekind if, 0),( MNMHom for all N se M. In fact a submodule N of M is called semiessential in M and denoted by (N se M ) if, 0 pN for all nonzero prime submodules P of M [1], provided that M has nonzero prime submodule. In this paper we present the basic properties of strongly essentially quasi-Dedekind and some relationships with other modules. Next throughout this paper, M has a nonzero prime submodules. 1.1 Definition An R-module M is called strongly essentially quasi-Dedekind if, 0),( MNMHom for all semiessential submodules N of M. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 1.2 Remarks and Examples: 1- It is clear that if M is a strongly essentially quasi-Dedekind R-module, then M is an essentially quasi-Dedekind R-module, since every essential submodule is semiessential submodule. 2- Every quasi-Dedekind R-module is a strongly essentially quasi-Dedekind R-module, but the converse is not true in general, for example: Z6 as Z- module is strongly essentially quasi-Dedekind, but it is not quasi-Dedekind, since 0)),2(( 266  ZZZHom . 3- Each of Z, Z6, Z10 is strongly essentially quasi-Dedekind as Z-module. 4- Each of Z4, Z8, Z12, Z16 is not strongly essentially quasi-Dedekind as Z-module. 5- p Z  is not strongly essentially quasi-Dedekind as Z-module ,for all prime numbers p. 6- 2ZZ  is not essentially quasi-Dedekind as Z-module, see [4, Remark 1.2.14], so it is not strongly essentially quasi-Dedekind as Z-module. 7- Let N  M and M/N is a strongly essentially quasi-Dedekind R-module, then it is not necessarily that M is a strongly essentially quasi-Dedekind R-module; For example : Let M = Z12 as Z-module and let N = 12(6) Z , then 12 6Z N Z is a strongly essentially quasi-Dedekind Z-module, but Z12 is not strongly essentially quasi-Dedekind as Z-module. Recall that a nonzero R-module M is called semi-uniform, if every nonzero R-submodule of M is a semiessential submodule of M [1]. 1.3 Proposition: Let M be a semi-uniform R-module. Then M is a quasi-Dedekind R-module if and only if M is a strongly essentially quasi-Dedekind R-module. Proof : It is clear. 1.4 Corollary: Let M be a uniform R-module .The following statements are equivalent: 1- M is a quasi-Dedekind R-module. 2- M is a strongly essentially quasi-Dedekind R-module. 3- M is an essentially quasi-Dedekind R-module. Proof : It is clear . The following is a characterization of strongly essentially quasi-Dedekind module. 1.5 Theorem: Let M be an R-module M is strongly essentially quasi-Dedekind if and only if for each )(MEndf R , 0f implies Kerf se M . Proof : ) Suppose that M is a strongly essentially quasi-Dedekind R-module .Let )(MEndf R , 0f . To prove that Kerf se M. Assume that Kerf se M, define MKerfMg : by g ( m+Kerf ) = f(m) for all mM. It is clear that g is well-defined and 0g , hence 0),( MKerfMHom which is a contradiction. ) Assume that there exists MNMh : , 0h , for some N se M. Consider the following : MNMM h , where  is the natural projective mapping, then مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 )(MEndho R   and 0 . Since N  Ker and N se M, thus Ker se M. Since (for any prime submodule P of M, PN  (0)). But this is contradiction. 1.6 Proposition: Let M be an R-module and let R = R/J, MannJ R , then M is a strongly essentially quasi-Dedekind R-module if and only if M is a strongly essentially quasi-Dedekind R -module. Proof : Since ),( MNMHom R = ),( MNMHomR for all N  M, by [6, p.51] the result follows easily. Recall that an injective R-module E( M) is called an injective hull ( injective envelope ) of an R-module M if, there exists a monomorphism f : M  E(M) such that Imf e E(M) [6, p.142]. And recall that a quasi-injective R-module M is called a quasi-injective hull (quasi-injective envelope) of an R-module M if, there exists a monomorphism g: M  M such that Img e M [11]. To prove the next result, we state and prove the following lemma: 1.7 Lemma: Let M be an R-module and let A  M, B  M. If A se B se M then A se M. Proof : Let P be a nonzero prime submodule in M, then 0  PB is prime in B and to show this: Let x B , r R. If rx  PB, then rx  P and rx  B. Now rx  P implies either x P or r [ P:M], since P is prime in M. If x P, then x  PB. And if r  [ P : M], then rM P, but rB  rM  P, then rB  P and also rB  B, hence rB  PB. Thus r [ PB:B], so that PB is prime in B. It follows that A (PB)  0 and hence AP  0. Therefore A se M. 1.8 Proposition: Let M be an R-module. If M is a strongly essentially quasi-Dedekind R-module, then M is a strongly essentially quasi-Dedekind R-module. Proof : Let )(MEndf R , f 0. To prove that Kerf se M. Since M is quasi- injective R-module, then there exists g : M  M , g  0 such that g o i = i o f ( where i is the inclusion mapping ). i f i M M g M M مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 But M is a strongly essentially quasi-Dedekind R-module, so Kerg se M , but Kerf  kerg, then Kerf  se M, and since M e M ; that is M se M , then by (Lemma 1.7) Kerf se M which implies Kerg se M which is a contradiction. Thus M is a strongly essentially quasi- Dedekind R-module. The following results follow directly by (Prop . 1.8). 1.9 Corollary: Let M be a strongly essentially quasi-Dedekind and quasi-injective R-module. If N se M then N is a strongly essentially quasi-Dedekind R-module. 1.10 Corollary: Let M be an R-module. If E(M) is a strongly essentially quasi-Dedekind R-module then M is a strongly essentially quasi-Dedekind R-module . The converse of (Coro. 1.10) is not true in general, as the following example shows: 1.11 Example: It is well known that Z2 as Z-module is a strongly essentially quasi-Dedekind. But E(Z2) = Z2 ∞ is not strongly essentially quasi-Dedekind as Z-module . 1.12 Remark: Let M be an R-module. If N  M is a strongly essentially quasi-Dedekind R-module then it is not necessarily that M/N is a strongly essentially quasi-Dedekind R-module, consider the following example: 1.13 Example: Let M = Z as Z-module, and let N = 4Z  Z = M. It is clear that N se M and M is strongly essentially quasi-Dedekind and quasi-injective as Z-module, so by (Coro.1.9) N is strongly essentially quasi-Dedekind as Z-module, but M/N = Z/4Z  Z4 is not strongly essentially quasi-Dedekind as Z-module ( see, Rem.and.Ex(1.2)(4)) . Recall that a nonzero R-module M is called compressible if, M embedded in each of its nonzero submodules [2]. 1.14 Proposition: Let M be a multiplication R-module, N ≨ M. If N is a prime R-submodule of M, then M/N is a strongly essentially quasi-Dedekind R-module. Proof: Since N is a prime submodule of M, so by [12, Coro. 4.18, ch.1] M/N is a compressible R-module, thus by [7,Prop 2.6, p.30] M/N is a quasi-Dedekind R-module. Therefore by (Rem.and.Ex(1.2)(2)) M/N is a strongly essentially quasi-Dedekind R-module. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 To prove our next result, we need the following lemma: 1.15 Lemma: Let M, N be an R-modules, let f : M  N be a monomorphism. Let K  M, A  N, then: (1) K se M implies f (K) se N. (2) A se N implies )(1 Af  se M, if f is an epimorphism and Kerf  P, where P is any prime submodule of M. Proof : (1) Suppose that there exists a nonzero prime submodule W of N such that f(K) W = 0. But K = ))((1 Kff  , since f is a monomorphism. Hence K  )(1 Wf  = ))((1 Kff   )(1 Wf  = ))((1 WKff  = )0(1f = Kerf = {0}. But )(1 Wf  is a nonzero prime submodule of M, so K ≰se M which is a contradiction. (2) The proof is similarly. 1.16 Proposition: Let M  N. Then M is a strongly essentially quasi-Dedekind R-module if and only if N is a strongly essentially quasi-Dedekind R-module. Proof :  ) Let  : M  N be an isomorphism. Suppose that M is a strongly essentially quasi-Dedekind R-module. Let fEndR(N), f  0 . To prove that Kerf ≰se N, consider the following: M  N  f N  1  M, let h= -1o f o EndR(M), h  0 since h(M) =  -1∘ f ∘ (M)   -1 (f(N) )   -1 (N)  0 .Then Kerh ≰se M, since M is a strongly essentially quasi-Dedekind R-module. We claim that Kerf = {yN: -1(y)  Kerh}. To prove our assertion. Let y Kerf , then f(y) = 0. h( -1(y))=  -1∘ f∘ ( -1(y))= -1∘ f (y)= -1 (0)=0. Thus for each y Kerf, then  -1(y)  Kerh and hence  -1(Kerf)Kerh ≰se M this implies  -1(Kerf) ≰se M, so by (Lemma.(1.16)(2)) Kerf ≰se N. Therefore N is a strongly essentially quasi-Dedekind R-module. ) The proof of the converse is similarly. 1.17 Theorem: Let M be an R-module such that M/V is projective R-module, for all V se M. If M is a strongly essentially quasi-Dedekind R-module, then M/N is a strongly essentially quasi- Dedekind R-module for all N M. Provided N se M. Proof: To prove that M/N is strongly essentially quasi-Dedekind, we must prove that ),( N M NU NM Hom = 0 for all U/N se M/N. By 3rd isomorphism theorem U M N U N M  , so its enough to show that Hom( M/U, M/N)= 0. Let f  Hom( M/U, M/N ), f  0 .Hence there exists g : M/U  M such that  og = f, since M/U is projective. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 So g  0, thus 0),( MUMHom . U se M, because NU  . Thus M is not strongly essentially quasi-Dedekind R-module, so we get a contradiction .Thus M/N must be a strongly essentially quasi-Dedekind R-module. To prove the next theorem we need the following lemma: 1.18 Lemma: Let M1, M2 be R-modules. If A se M1, B se M2 then BA se 21 MM  . Proof : Let P be prime in 21 MM  , then by [5] P = 21 PP  , such that either p1, p2 prime in M1 , M2 respectively , so 0)()()()( 2121  PBPAPPBA . Or , P = 21 MP  , then 0)()()()( 12121  BPAMBPAMPBA . Or , P = 21 PM  , then 0)()()()( 22121  PBAPBMAPMBA . 1.19 Theorem: A direct summand of a strongly essentially quasi-Dedekind R-module is a strongly essentially quasi-Dedekind R-module. Proof : Let 21 MMM  . To prove M1 is a strongly essentially quasi-Dedekind R-module. Let 0),( 1  fMEndf R , we have the following diagram: 211121 MMMMMM if   R i f End (M)o o .If 1i f (M) i f(M )  o o o 0)())(( 11  MfMfi ,then Ker(i f )o o ≰seM. Ker(i f )o o ={m1+m2: }0),( 21 mmiofo ={m1+m2: 1i f (m ) 0}o ={m1+m2: }0)( 1 mf = 2MKerf  ≰se 21 MM  . But M2se M2, so Kerf ≰se 1 M , by (Lemma 1.18). The converse of ( Theorem 1.19) is not true in general ,consider the following example: 1.20 Example: We know that each of Z, Z6 as Z-module is strongly essentially quasi-Dedekind. But 6Z Z is not strongly essentially quasi-Dedekind as Z-module, since 6Z Z is not essentially quasi-Dedekind. M U M N M 0 g  f مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Recall that a nonzero submodule N of an R-module M is called quasi-invertible if Hom(MN,M ) = 0, [7]. 1.21 Proposition: If M be a strongly essentially quasi-Dedekind R-module. Then NannMann RR  for all N se M. Proof : Suppose that M is a strongly essentially quasi-Dedekind R-module, then 0),( MNMHom for all N se M, hence N is a quasi-invertible submodule of M, for all N se M . Thus by [7, Prop .1.4] NannMann RR  for all N se M. To prove the following proposition, we need to prove the following lemma: 1.22 Lemma: Let M be a faithful multiplication R-module. Then N se M if and only if [N :M] se R. Proof :  ) If N se M. Let P be any nonzero prime ideal in R. Then by [3, Lemma 2.10] PM is a nonzero prime submodul in M, hence N PM 0; that is [(N:M)M ]PM 0, and since M is a faithful multiplication R-module, [ (N:M) P] M  0, by [3]. Thus [N:M] P  0, so [N:M] se R . ) If [N :M] se R .Let P be any nonzero prime submodule in M, then by [3, Prop .2.8,ch1] [P:M] is prime ideal in R, and since [N :M] se R, we have [N:M]  [P:M]  0 which implies ( [N:M]  [P:M])M  0, so that by [3] [N:M]M [P:M] M  0, thus N P  0; that is N se M. 1.23 Proposition: Let M be a faithful multiplication R-module. If M is a strongly essentially quasi-Dedekind R-module, then R is a strongly essentially quasi-Dedekind R-module. Proof : Let f : R  R , f  0. For any rR, f (r) = r f(1) = ra , where a = f(1). Define g: M  M by g(m) = am for each mM. g is well-defined and g  0 , hence Kerg ≰se M. But Kerg = [Kerg:M]M, since M is a multiplication R-module. However we can show that [Kerg:M] = Kerf as the following: Let r [Kerg:M] implies rM Kerg, so g(rM) = 0, hence arM = 0; that is ar annRM = 0, thus f(r) = ar = 0, hence r Kerf. Now, let rKerf, then ar = f(r) = 0, so arM = 0; that is g(rM) = 0, thus rM Kerg and hence r [Kerg:M]. Therefore [Kerg:M] = Kerf. But Kerg ≰se M, implies by (Lemma (1.22)) [ Kerg:M ] ≰ se R, thus Kerf ≰se R and hence R is a strongly essentially quasi-Dedekind R-module. Recall that an R-module M is called scalar if for each f  EndR(M) , there exists rR such that f(a) = ar for all aM [10, p.8]. 1.24 Proposition: Let M be a finitely generated faithful multiplication R-module. If R is a strongly essentially quasi-Dedekind R-module, then M is a strongly essentially quasi-Dedekind R-module. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Proof : Since M is a finitely generated multiplication R-module, then by [9,Th.2.3] M is a scalar R-module, so for each f EndR(M), there exists rR such that f(m) = rm, for all mM. Define g : R  R by g(a) = ra, for all aR , Kerg ≰se R, since R is a strongly essentially quasi-Dedekind R-module. But Kerf = [Kerf: M] M, also by the same argument of the proof of (Prop .1.23), we get Kerg = [Kerf: M], but Kerg ≰se R, so [Kerf:M] ≰se R which implies Kerf ≰se M, by (Lemma 1.22). Thus M is a strongly essentially quasi-Dedekind R-module. By combining (Prop 1.23) and (Prop 1.24) , we get the following result: 1.25 Corollary: Let M be a finitely generated faithful multiplication R-module.M is a strongly essentially quasi-Dedekind R-module if and only if R is a strongly essentially quasi-Dedekind R-module. We end this paper with the following corollary: 1.26 Corollary: Let M be a finitely generated faithful multiplication R-module. If R is a strongly essentially quasi-Dedekind R-module, then EndR(M) is a strongly essentially quasi-Dedekind ring . Proof : Since M is a finitely generated multiplication R-module , then by [9,I.2.3] M is a scalar R-module. Then by [8, Lemma 6.2, ch.3] EndR(M)  R/annRM  R, but R is a strongly essentially quasi-Dedekind ring ,thus by Prop .1.16 EndR(M) is a strongly essentially quasi- Dedekind ring. References 1. AL-Daban,N.K.(2005)Semi-Essential Submodules and Semi-Uniform Modules, M.Sc.Thesis, College of Education, University of Tikret. 2. Desale, G.and Nicholson, W.K. (1981) Endoprimitive Rings, J.Algebra, 70: 548-560. 3. EL-Bast, Z.A.and Smith P.F., (1988), Multiplication Modules, Comm. In Algebra, 16: 755 – 779. 4. Ghawi, I.Y. (2010), Some Generalizations of Quasi-Dedekind Modules, M.Sc. Thesis, College of Education Ibn L-Haitham,University of Baghdad. 5. Hadi, I.M-A. (2003), Piecewise Noetherian Modules, Ph.D.Thesis, College of Education Ibn AL-Haitham,University of Baghdad. 6. Kasch, F. (1982), M odules and Rings, Academic Press, London. 7. Mijbass, A.S. (1997), Quasi-Dedekind Modules, Ph.D.Thesis, College of Science, University of Baghdad. 8. Mohamed–Ali, E.A. (2006), On Ikeda-Nakayama Modules, Ph.D.Thesis, College of Education Ibn AL-Haitham, University of Baghdad. 9. Naoum,A.G. (1990), On the Ring of Endomorphisms of a Finitely Generated Multiplication Modules, Periodica Math, Hungarica, 21(3):249 – 255. 10. Shihab B.N. (2004), Scalar Reflexive Modules, Ph.D.Thesis, College of Education Ibn AL-Haitham, University of Baghdad. 11. 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In Algebra,.14 (6):1141 – 1169. ،رسـالة ماجـستیر، حول المودیـوالت الجزئیـة األولیـة والمودیـوالت الجزئیـة شـبھ االولیـة،)1996 (،یمان علي عذابإ .12 .جامعة بغداد، كلیة العلوم مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 بقوة دیدیكاندیة الواسعة-المقاسات شبھ ثائر یونس غاوي، هاديإنعام محمد علي جامعة بغداد، ابن الهیثم -لیة التربیةك ،قسم الریاضیات جامعة القادسیة،كلیة التربیة ،قسم الریاضیات 2011تموز 13: ، قبل البحث في 2011 نیسان 5: استلم البحث في الخالصة واسـعة دیدیكاندیـة ال-ّفـي هـذا البحـث قـدمنا ودرسـنا مفهـوم المقاسـات شـبه . عنـصر محایـد ا حلقة أبدالیـة ذRلتكن دیدیكانـدي واسـع بقـوة إذا -ًمقاسـا شـبهR علـى Mّ یـسمى المقـاس اذ ، دیدیكاندیـة الواسـعة-بقوة كأعمـام إلـى المقاسـات شـبه ),(0كـان MNMHom لكـل مقـاس جزئـي شـبه واسـع N فـيM . یطلـق علـى مقـاس جزئـيN مـن مقـاس MعلـىR 0 شــبه واسـع إذا كــان pN لكــل مقــاس جزئــي أولــي غیـر صــفري P فــي M . علــى شــرط انM ـا لهـ . صفریةمقاسات أولیة غیر وة، المقاسـات الجزئیـة شـبھ ة المقاسـات شـبھ الدیدكاندیـھ الواسـعة، المقاسـات شـبھ الدیدكاندیـ:الكلمات المفتاحیة الواسـعة بـق .الواسعة، المقاسات الجدائیة مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012