ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Strongly Essentially Quasi-Dedekind Modules I. M.A.Hadi, T.Y.Gha wi Department of Mathematics, College of Education I bn-Al-Haitham , Unive rsity of Baghdad Department of Mathematics, College of Education, Unive rsity of Al- Qadisiya Received in: 5 April 2011, Accepted in: 13July 2011 Abstract Let R be a commutative ring with unity . In this p aper we introduce and study the concep t of st rongly essentially quasi-Dedekind module as a generalization of essentially quasi- Dedekind module. A unitary R-module M is called a st rongly 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 p rime submodules P of M . Key Words: Essentially quasi-Dedekind M odules; Strongly essentially quasi-Dedekind M odules, Semiessential submodules, M ultip lication M odules. 1. Introduction Let R be a commutative ring with unity and M be an R-module. M ijbass A.S in [7] introduced and st udied 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 st udied 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 p aper we give a generalization of essentially quasi-Dedekind which we call it st rongly 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 p rime submodules P of M [1], p rovided that M has nonzero p rime submodule. In this p aper we p resent t he basic prop erties of st rongly essentially quasi-Dedekind and some relationships with other modules. Next throughout this p aper, M has a nonzero p rime submodules. 1.1 De fini tion An R-module M is called strongly essentially quasi-Dedekind if, 0),( MNMHom for all semiessential submodules N of M. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 1.2 Remarks and Exam ples: 1- It is clear that if M is a st rongly 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 st rongly essentially quasi-Dedekind R-module, but the converse is not true in general, for examp le: Z6 as Z- module is st rongly essentially quasi-Dedekind, but it is not quasi-Dedekind, since 0)),2(( 266  ZZZHom . 3- Each of Z, Z6, Z10 is st rongly essentially quasi-Dedekind as Z-module. 4- Each of Z4, Z8, Z12, Z16 is not st rongly essentially quasi-Dedekind as Z-module. 5- p Z  is not st rongly essentially quasi-Dedekind as Z-module ,for all p rime numbers p. 6- 2ZZ  is not essentially quasi-Dedekind as Z-module, see [4, Remark 1.2.14], so it is not st rongly essentially quasi-Dedekind as Z-module. 7- Let N  M and M /N is a st rongly essentially quasi-Dedekind R-module, then it is not necessarily that M is a strongly essentially quasi-Dedekind R-module; For examp le : 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 st rongly 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 Proposi tion: 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 st atements 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 The orem: Let M be an R-module M is st rongly essentially quasi-Dedekind if and only if for each )(MEndf R , 0f imp lies Kerf se M . Proof : ) Sup p ose that M is a st rongly essentially quasi-Dedekind R-module .Let )(MEndf R , 0f . To p rove 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 p rojective map p ing, then ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 p rime submodule P of M , PN  (0)). But this is contradiction. 1.6 Proposi tion: Let M be an R-module and let R = R/J, MannJ R , then M is a st rongly essentially quasi-Dedekind R-module if and only if M is a st rongly essentially quasi-Dedekind R -module. Proof : Since ),( MNMHom R = ),( MNMHom R 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 monomorp hism 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 monomorp hism g: M  M such that Img e M [11]. To p rove the next result, we state and p rove 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 p rime submodule in M, then 0  P  B is p rime in B and to show this: Let x B , r R. If rx  P  B, then rx  P and rx  B. Now rx  P imp lies either x P or r [ P:M], since P is p rime 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 p rime in B. It follows that A  (P  B)  0 and hence A  P  0. Therefore A se M. 1.8 Proposi tion: Let M be an R-module. If M is a st rongly essentially quasi-Dedekind R-module, then M is a strongly essentially quasi-Dedekind R-module. Proof : Let )(MEndf R  , f  0. To p rove 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 mapp ing ). i f i M M g M M ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 But M is a st rongly 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 imp lies 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 st rongly 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 examp le shows: 1.11 Example: It is well known that Z2 as Z-module is a st rongly essentially quasi-Dedekind. But E(Z2) = Z2 ∞ is not st rongly 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 st rongly essentially quasi-Dedekind R-module, consider the following examp le: 1.13 Exam ple: Let M = Z as Z-module, and let N = 4Z  Z = M. It is clear that N se M and M is st rongly essentially quasi-Dedekind and quasi-injective as Z-module, so by (Coro.1.9) N is st rongly essentially quasi-Dedekind as Z-module, but M/N = Z/4Z  Z4 is not st rongly essentially quasi-Dedekind as Z-module ( see, Rem.and.Ex(1.2)(4)) . Recall that a nonzero R-module M is called comp ressible if, M embedded in each of its nonzero submodules [2]. 1.14 Proposi tion: Let M be a multiplication R-module, N ≨ M. If N is a p rime R-submodule of M, then M/N is a strongly essentially quasi-Dedekind R-module. Proof: Since N is a p rime 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. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 To p rove our next result, we need the following lemma: 1.15 Lemma: Let M, N be an R-modules, let f : M  N be a monomorp hism. Let K  M, A  N, then: (1) K se M imp lies f (K) se N. (2) A se N imp lies )( 1 Af  se M, if f is an epimorp hism and Kerf  P, where P is any p rime submodule of M. Proof : (1) Sup p ose 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 p rime submodule of M, so K ≰se M which is a contradiction. (2) The proof is similarly. 1.16 Proposi tion: 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. Sup p ose that M is a st rongly essentially quasi-Dedekind R-module. Let fEndR(N), f  0 . To p rove that Kerf ≰se N, consider the following: M  N  f N  1  M, let h= -1 o 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 st rongly essentially quasi-Dedekind R-module. We claim that Kerf = {yN: -1 (y)  Kerh}. To p rove 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 imp lies  -1 (Kerf) ≰se M, so by (Lemma.(1.16)(2)) Kerf ≰se N. Therefore N is a st rongly essentially quasi-Dedekind R-module. ) The proof of the converse is similarly . 1.17 The orem: Let M be an R-module such that M/V is p rojective R-module, for all V se M. If M is a st rongly essentially quasi-Dedekind R-module, then M/N is a st rongly essentially quasi- Dedekind R-module for all N M. Provided N se M. Proof: To p rove that M/N is st rongly essentially quasi-Dedekind, we must p rove that ),( N M NU NM Hom = 0 for all U/N se M/N. By 3 rd 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 p rojective. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 st rongly essentially quasi-Dedekind R-module, so we get a contradiction .Thus M/N must be a st rongly essentially quasi-Dedekind R-module. To p rove 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 p rime in 21 MM  , then by [5] P = 21 PP  , such that either p1, p2 p rime in M1 , M2 resp ectively , so 0)()()()( 2121  PBPAPPBA . Or , P = 21 MP  , then 0)()()()( 12121  BPAMBPAMPBA . Or , P = 21 PM  , then 0)()()()( 22121  PBAPBMAPMBA . 1.19 The orem: A direct summand of a st rongly essentially quasi-Dedekind R-module is a st rongly essentially quasi-Dedekind R-module. Proof : Let 21 MMM  . To p rove 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 = 2 MKerf  ≰se 21 MM  . But M2 se M2, so Kerf ≰se 1M , by (Lemma 1.18). The converse of ( Theorem 1.19) is not true in general ,consider the following examp le: 1.20 Example: We know that each of Z, Z6 as Z-module is st rongly essentially quasi-Dedekind. But 6Z Z is not st rongly essentially quasi-Dedekind as Z-module, since 6Z Z is not essentially quasi-Dedekind. M U M N M 0 g  f ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 Proposi tion: If M be a st rongly essentially quasi-Dedekind R-module. Then NannMann RR  for all N se M. Proof : Sup p ose that M is a st rongly 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 p rove the following p rop osition, we need to p rove the following lemma: 1.22 Lemma: Let M be a faithful multip lication 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 p rime 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 p rime submodule in M, then by [3, Prop .2.8,ch1] [P:M] is p rime ideal in R, and since [N :M] se R, we have [N:M]  [P:M]  0 which imp lies ( [N:M]  [P:M])M  0, so t hat by [3] [N:M]M  [P:M] M  0, thus N  P  0; that is N se M. 1.23 Proposi tion: 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] imp lies 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, imp lies 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 Proposi tion: Let M be a finitely generated faithful multiplication R-module. If R is a st rongly essentially quasi-Dedekind R-module, then M is a st rongly essentially quasi-Dedekind R-module. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 imp lies 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 t he 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 p aper with t he following corollary : 1.26 Corollary: Let M be a finitely generated faithful multiplication R-module. If R is a st rongly essentially quasi-Dedekind R-module, then EndR(M) is a st rongly 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. 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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 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 لهــا . صفریةمقاسات أولیة غیر وة، المقاسـات الجزئیـة شـبھ ة المقاسـات شـبھ الدیدكاندیـھ الواسـعة، المقاسـات شـبھ الدیدكاندیـ:الكلمات المفتاحیة الواسـعة بـق .الواسعة، المقاسات الجدائیة ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012