IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Annsemimaximal and Coannsemimaximal Modules I. M.A.Hadi, H. Y. Khalaf Departme nt of Mathematics, College of Education Ibn-Al-Haitham Unive rsity of Baghdad Received in : 20 September 2010 Accepte d in : 8 February 2011 Abstract Some authors st udied modules with annihilator of every nonzero submodule is p rime, p rimary or maximal. In this p aper, we introduce and study annsemimaximal and coannsemimaximal modules, where an R-module M is called annse mimaximal (resp . coannsemimaximal) if annRN (resp . R M ann N ) is semimaximal ideal of R for each nonzero submodule N of M . Keywords: Annsemima ximal modu le, semisimp le module, semisi mple ring, semip rime module, ma x-module, uniform module, Z-re gular module, F-regular module, Artinian module, flat module, cop rime module, coannsemimaximal module. Introduction Let R be a commutative ring with unity and let M be an R-module. M untaha A.R.H. in [1] introduced and st udied quasi-prime modules where an R-module M is quasi-prime if annRN is a p rime ideal of R for every nonzero submodule N of M . Adwia J.A.A. in [2] introduced and st udied quasi-primary modules, where an R-module M is called quasi-primary if annRN is a p rimary ideal of R, for each nonzero submodule N of M . Adwia J.A.A. in [3] introduced and st udied max modules, where an R-module M is said to be max module if annRN is a maximal ideal of R, for each nonzero submodule N of M . 1. Recall that an ideal I of R is called semimaximal if I is an intersection of finitely many maximal ideals of R, [4]. 2. In this p aper, we introduced and st udied annsemimaximal and coannsemimaximal modules where an R-module M is called annsemimaximal (resp . coannsemimaximal) if annRN (resp . R M ann N ) is a semimaximal ideal of R. 1- Annse mimaximal Modul es In this section, we introduce the concept of annsemimaximal modules. We give some characterizations t o this concept and est ablish some basic prop erties of this concept. 1.1 De fini tion: Let M be an R-module. M is called annsemimaximal module if annRN is a semimaximal ideal of R for each non-zero submodule N of M . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 1.2 Remarks and Examples: (1) p Z  is not annsemimaximal Z-module. (2) Z6 as a Z-module is annsemimaximal module. (3) Z as a Z-module is not annsemimaximal module. (4) Q as a Z-module is not annsemimaximal module. (5) Zp as a Z-module is annsemimaximal module. (6) for each nZ+, ZZn is not annsemimaximal Z-module. (7) Every submodule N of an R-module M (where M is annsemimaximal module) is annsemimaximal module. Proof: Let K be a nonzero submodule of N. Then K be a non-zero submodule of M and so that annRK is semimaximal ideal (since M is annsemimaximal module). (8) Let M be annsemimaximal module and let N  M . Then   is annsemimaximal module. Proof: Let :MM /N be the natural epimorp hism and M is annsemimaximal module. Then for each non-zero submodule W of M , annRW is semimaximal ideal of R. But annRWannRW/N. Hence annRW/N is semimaximal ideal by [5,p rp .(1.2.11)]. Thus   is annsemimaximal module. (9) The homomorphic image of annsemimaximal module is annsemimaximal module. Proof: Let f:MM ' be an ep imorp hism such that M is annsemimaximal module. Then by the first fundamental theorem of homomorp hisim, M ' ker f   . But ker f  is annsemimaximal by (8). Hence M ' is annsemimaximal module. Now, we have the following characterization of annsemimaximal module. 1.3 Proposi tion: Let M be an R-module. Then M is annsemimaximal module if and only if annRM is a semimaximal ideal of R. Proof: () It follows directly by definition (1.1). () let (0)  N be a submodule of M . Then annRN  annRM . But annRM is semimaximal, so by [5,p rop . (1.2.11)], annRN issemimaximal. Thus M is annsemimaximal module. 1.4 Corollary: An R-module M is annsemimaximal if and only if R/annRM is semisimple ring. Proof: By p rop osition (1.3) M is annsemimaximal module  annRM is a semimaximal ideal.  R/annRM is semisimple ring. Now, we have the following theorem. 1.5 The orem: Let M be an R-module. Then (1)  (2), (2)  (3), (3)  (4), (4)  (1) if M is finitely generated (1) M is annsemimaximal module. (2) [annRN R : A] is a semimaximal for each non-zero submodule N of M and for each ideal A of R such that A  annRN. . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 [annN R : r] is a semimaximal ideal of R for each non-zero submodule N of M , rR such that (r)  annRN. (3) AnnR(m) is a semimaximal ideal of R for each m0, mM . Proof: (1)  (2), sup p ose that M is annsemimaximal module. Let N be a non-zero submodule of M . Then annRN is semimaximal ideal of R. Assume that A is an ideal of R such that A  annRN. It is clear that annRN  [annRN R : A]. So, according to [5,coro.(1.2.12)], [annRN R : A] is semimaximal ideal of R. (2)  (3), take A = (r) the ideal of R generated by r, the result follows by (2). (3)  (4), let 0m M . Because 1annR(m). [annR(m):R] is semimaximal ideal of R by (3). But [annR(m):R]=annR(m). So annR(m) is semimaximal ideal of R. (4)  (1), since M is finitely generated, M = n i i 1 Rx   , xiM , annRM = n i i 1 annx   . But ann(xi) for all i=1,…,n is semimaximal ideal. So, by [5,coro.(1.2.15)], annRM is semimaximal. Thus M is ansemimaximal by p rop . (1.3). Recall that an R-module M is called semisimple if every submodule of M is a direct summand of M . And a ring R is said to be semisimple ring if and only if R is a semisimp le R- module, [6]. 1.6 Proposi tion: Every semisimple R-module M is annsemimaximal. Proof: By [6,p rop .(1.1.46)], we get R/annRM is a semisimple ring. Therefore annRM is a semimaximal ideal by [4,p rop .(1.2.5)]. Thus M is annsemimaximal by p rop . (1.3). The following corollary is an ap p lication of prop osition (1.6). 1.7 Corollary: Let R be a semisimple ring. Then every R-module M is annsemimaximal. Proof: It is known that if R is semisimple ring, then M is semisimple module [5,p rop .(1.1.44)]. Hence M is annsemimaximal module by p revious p rop osition. Next, we have the following p rop osition. 1.8 Proposi tion: If M is an Artinian and annsemimaximal R-module, then M is semisimple. Proof: We have M is annseimmaximal, then annRM is semimaximal. Thus J(M )= 0 by [5,coro,(1.3.6)]. But M is an artinian and J(M )=0, then M is semisimple, [5]. 1.9 Example: Z24 as a Z-module is not annsemimaximal module and Z24 is not semisimple. The following result is consequence of p rop osition (1.8). 1.10 Corollary: Let M is an Art inian R-module.Then M is semisimple module if and only if M is annsemimaximal. 1.11 Proposi tion: If M is annsemimaximal R-module, then every cyclic submodule of M is semisimple. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Proof: If M is annsemimaximal R-module, then annR(x) is semimaximal ideal by Th.((1.5),(4)), so by [5,p rop .(2.3.15)], we get the result Now, we induced the following corollary . 1.12 Corollary: If M is finitely generated and annsemimaximal R-module, then M is semisimple R-module. Proof: Let M =Rx1 + Rx2 + … + Rxn for some x1, x2, …, xn. But Rxi is semisimple by p revious p rop osition. Therefore M = n i i 1 Rx   is semisimple By combining corollary (1.12), p rop osition (1.6), we get t he following: 1.13 Corollary: Let M be a finitely generated R-module. Then M is annsemimaximal module if and only if M is semisimple. 1.14 Corollary: R is a semisimp le ring if and only if R is annsemimaximal ring. Now, we turn our att ention to direct sum of annsemimaximal modules. 1.15 Proposi tion: Let M be a faithful R-module. Then R is semisimple if and only if M is annsemimaximal. Proof: () directly from [5,prop .(1.1.44)] and prop osition (1.6). () if M is annsemimaxi, then annRM is a semimaximal ideal; that is (0) is a semimaximal ideal. Thus R/(0) � R is semisimple. By combining corollary (1.13), p rop osition (1.15) and corollary (1.14), we get the following: 1.16 Corollary: Let M be a faithful finitely generated R-module. The following st atements are equivalent: (1) M is annsemimaximal. (2) M is semisimple. (3) R is semisimple. (4) R is annsemimaximal. Now, we give the following p rop osition. 1.17 Proposi tion: If R is a local ring and M is annsemimaximal R-module, then M is semisimple.. Proof: M is annsemimaximal module. Then annRM is semimaximal ideal. Thus the result follows by [5,coro.(1.3.7)]. 1.18 Proposi tion: Let M 1, M 2 be two R-modules, M =M 1M 2. Then M is annsemimaximal if and only if M 1, M 2 are annsemimaximal R-module. Proof: () let 1:M  M 1, 2:M  M 2 be the natural projections. T hus M 1 and M 2 are annsemimaximal modules by remarks and examp les ((1.2),(9)). () we have annRM 1 is semimaximal ideal and annRM 2 is semimaximal by p rop osition (1.3). On t he other hand annR(M 1M 2) = annRM 1 annRM 2. But by [5, coro.(1.2.15)], IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 annRM 1 annRM 2 is semimaximal. Therefore annR(M 1M 2) is semimaximal. Thus M 1M 2 is annsemima ximal modu le, by p rop .(1.3). .. Recall that an R-module M is called semip rime if and only if annRN is a semiprime ideal of R, for each non-zero R-submodule N of M , [7,Def.(4.1.1)]. By using this concept, we have the following. 1.19 Proposi tion: Every annsemimaximal R-module is semip rime R-module. Proof: Let M be an annsemimaximal module. Then for each non-zero submodule N of M , annRN is semimaximal ideal of R. Thus by [5,p rop .(1.2.21)], annRN is semip rime and hence M is a semiprime module. The converse of this p rop osition is not true in general. For examp le:Z as a Z-module is semip rime module, but it is not annsemimaximal module by remarks and examp les ((1.2),(3)). For our next corollary the following definitions are needed. An R-module M is said to be serial (chain) R-module if the R-submodules of M are linearly orderd with resp ect to inclusion, [6], [7]. An R-module M is said to be a p rime module if annRM =annRN for every non-zero submodule N of M , [8], [9]. As an ap p lication of prop osition (1.19), we give the following corollary . 1.20 Corollary: Let M be a serial annsemimaximal module. Then M is p rime R-module. Proof: From p rop osition (1.19), M is semip rime module and from [7,p rop .(4.2.1)], we get the result. Recall that an R-module M is said to be a max-module if Rann N is maximal ideal of R for each non-zero submodule N of M , [3]. In the class of max-module. The two concept of annsemimaximal module and semip rime module are equivalent. 1.21 Proposi tion: Let M be a max-module. Then M is annsemimaximal module if and only if M is semip rime module. Proof: Sup p ose that M is semip rime R-module. Then for each a non-zero submodule N of M , annRN is semiprime ideal of R, that is annRN= Rann N for each non-zero submodule N of M . But M is max-module which imp lies that Rann N is maximal ideal of R for each non-zero submodule N of M and hence annRN is maximal ideal for each non-zero submodule N of M by [5,Rem.(1.2.2),(2)], annRN is semimaximal ideal of R and hence M is annsemimaximal module. Conversely : It follows by p rop osition (1.19). Now, the following results are other consequences of p rop osition (1.21), but first we need to recall some definitions. An R-module M is called Z-regular module if for all mM , there exists fHomR(M ,R)=M * such that f(m)m = m, [10]. An R-submodule N of M is called essential in M if for each non-zero R-submodule L of M , NL0, [6].And an R-module M is called uniform if every non-zero R-submodule of M is essential. An R-submodule N of M is called quasi-invertible if Hom(   ,M )=0. And an R-module M is called quasi-Dedekind if every non-zero R-submodule of M is quasi-invertible, [11]. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Hence, we have the following consequences of (1.21). 1.22 Corollary: If M is max-module and Z-regular module. Thus M is annsemimaximal module. Proof: It follows directly from prop osition (1.21) and [7,p rop .(4.2.2)]. 1.23 Corollary: Let M be a uniform annsemimaximal R-module. Then M is quasi-Dedekind. Proof: M is annsemimaximal module, then M is semip rime by p rop osition (1.21) and by [7,p rop .(4.2.4)], we get t he result. Now, we can give the following p rop osition. 1.24 Proposi tion: Let M be a uniform max-R-module. Then the following st atements are equivalent. (1) M is annsemimaximal module. (2) M is seiprime module. (3) M is quasi-Dedekind. (4) M is p rime. Proof: (1)  (2) by p rop osition (1.19). (2)  (3) by [7,p rop .(4.2.4)]. (3)  (4) by [11,p rop .(1.7), ch.2]. (4)  (1) It is clear that every p rime module is semip rime module and hence by p rop osition (1.21) we get t he result. Recall that an R-module M is said to be regular module if R/annR(x) is regular ring for all 0  x  M , [5]. By using this concept, we have the following. 1.25 Remark: Every annsemimaximal module is regular module. Proof: Let M be annsemimaximal R-module. Then annRM is semimaximal ideal and by [5,p rop .(1.3.5)], M is regular module. 1.26 Proposi tion: If M is annsemimaximal R-module, then M /N is regular R-module for all submodules N of M . Proof: Let M is annsemimaximal module. Then annRM is semimaximal. But annRM  [N R : M ] for all submodule N of M , so [N R : M ] is semimaximal ideal by [5,p rop .(1.2.11)]. Hence M /N is regular R-module by [5,p rop .(1.3.8)]. The Jacobson radical of an R-module M denoted by J(M ), is defined to be the intersection of all maximal submodules of M , in case M has maximal submodules and J(M )=M in case M has no maximal submodule, [6]. 1.27 Remark: Let M be an annsemimaximal R-module. Then J(M )=0. Proof: It is abvious according to [5,coro.(1.3.6)]. Recall that an R-module M is called F-regular if every submodule of M is p ure [12,ch.2]. By using this concept, we give the following p rop osition. 1.28 Proposi tion: If M is annsemimaximal R-module, then M is F-regular. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Proof: We have M is annsemimaximal, then annRM is semimaximal. Thus every cyclic submodule is p ure by [5,p rop .(1.3.9)]. Hence M is F-regular. 1.29 Proposi tion: Let R be a PID, annRM0, M is p rime R-module. Then M is annsemimaximal module. Proof: Since M is p rime R-module. Then annRM is p rime ideal which imp lies that annRM is maximal ideal (since R is PID). Thus annRM is semimaximal ideal of R. Hence M is annsemimaximal R-module, by p rop osition (1.3). The converse of p rop osition (1.29) is not true, for examp le:Z6 as Z-module is annsemimaximal. But M is not p rime. Recall that an R-module M is flat if for each injective homomorp hisim f: N'  N from one R-module into another, the homomorphisim 1Mf:M R  N' M R  N is injective, where 1M is the identity isomorphisim of M , [6]. 1.30 Proposi tion: If M is flat annsemimaximal R-module, then every homomorp hic image of M is flat. Proof: We have M is annsemimaximal, then annRM is semimaximal ideal. Thus by [5,p rop .(1.3.10)], we get t he result. Next, we introduce the following definition. 1.31 De fini tion: Let N be a p rop er submodule of an R-module M . N is called quasi-semimaximal if [N R : (m)] is a semimaximal ideal for each m  N. 1.32 Remark: Let M be a finitely generated R-module, N be semimaximal submodule of M . Then [N R : M ] is semimaximal ideal. 1.33 Remark: Let M be a finitely generated R-module, N be semimaximal submodule of M . Then N is quasi-semimaximal submodule. Proof: By remark (1.32), [N R : M ] is semimaximal ideal. But for each mN, [N R : (m)]  [N R : M ]. Thus by [5,p rop .(1.2.11)], we get [N R : (m)] is semimaximal ideal of R. We end this section by the following result. 1.34 Proposi tion: Let M be a finitely generated R-module. Then M is annsemimaximal module if and only if (0) is quasi-semimaximal submodule of M . Proof: Sup p ose that M is annsemimaximal module. Then annR(m) is semimaximal ideal for each mM . Thus [0 R : m] is semimaximal ideal for each mM . Hence (0) is semimaximal ideal. Conversely : if (0) is quasi-semimaximal submodule of M , then [0 R : m] is semimaximal ideal for each mM . Therefore M is annsemmaximal by theorem ((1.5),(4)). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 2- Coannsemimaximal Modul es: In this section, we introduce the concept of coannsemimaximal module which is st rongly from the concept of annsemimaximal module in section one. We give some characterizations about t his concept and many results are st udied. We st art with t he following definition. 2.1 De fini tion: An R-module M is called co annsemimaximal module if annR M N is semimaximal id eal of R for each non-zero p rop er submodule N of M . Equivalently , M is coannsemimaximal if M N is annsemima ximal modu le for each non-zero prop er submodule N of M . 2.2 Examples: (1) Z12 is not coannsemimaximal Z-module, since if N = < 4 >, then annZ 12 Z N = annZZ4 = 4Z which is not semimaximal ideal. (2) 2pZ as a Z-module is coannsemimaximal, where p is a prime number. Proof: Since < p > is only non-zero p rop er submodule of 2pZ . 2pZ /< p > � Zp and annZZp=p Z which is clear semimaximal ideal. Next, we have the following p rop osition. 2.3 Proposi tion: Let M be an R-module. Then every annsemimaximal module is coannsemimaximal module. Proof: Let M be an annsemimaximal module. Then M N is annsemimaximal module by remarks and examp les ((1.2),(9)). Thus annR M N is semimaximal which imp lies that M is coannsemimaximal module. The converse of p rop osition (2.3) is not true in general. For examp le: Let Z9 be a Z- module. Then Z9 is coannsemimaximal module but not annsemimaximal. And Z4 as a Z- module is coannsemimaximal module but it is not annsemimaximal module. The following p rop osition p roves that the converse of (2.3) is true under the condition that M is coprime module, but first we need to recall the definition of coprime module. An R-module M is called coprime module if annRM = annR M N for every p rop er submodule N of M , [13]. 2.4 Proposi tion: Let M be a coprime and coannsemimaximal R-module. Then M is annsemimaximal R-module. Proof: Since M is coprime module. Then annRM =annR M N for every p rop er submodule N of M . But annR M N is semimaximal ideal of R for each non-zero p rop er submodule N of M (since M is coannsemimaximal). Thus annnRM is semimaximal ideal of R and hence M is annsemimaximal module by p rop osition (1.3). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 As an ap p lication of (2.4), we have the following. 2.5 Corollary: Let M be a coannsemimaximal R-module and M is a coprime E-module, where E=EndR(M ). Then M is annsemimaximal R-module. Proof: M is coprime E-module, then M is coprime R-module by [14,coro.(2.2.3)] and from p rop osition (2.4), we get t he result. . Recall that a non-simple R-module M is called antihop fian if MM /N for all p rop er submodules N of M , [15]. By using this concept we get t he following. 2.6 Proposi tion: Let M be an antihop fian, N is semimaximal submodule of M . Then M is coannsemimaximal R-module. Proof: We have N is semimaximal submodule. Then M N is semisimple by [5,Def.(2.1.1)]. Thus M N is annsemimaximal module by p rop .(1.5). But M M W N � for each p rop er submodule W of M , since M is antihop fian. That means M M N . Thus M W is annsemimaximal for all p rop er submodule W of M . Therefore M is coannsemimaximal module. Now, we p rove the following lemma. 2.7 Lemma: Let M be an R-module. If N is a semimaximal submodule, then [N R : M ] is semimaximal ideal. Proof: Sup p ose that N is a semimaximal submodule. Then by [5,def.(2.1.1)], M N is semisimple R-module and hence by p rop osition (1.6), M N is annsemimaximal module. Then by p rop osition (1.3), annR M N is semimaximal ideal. But [N R : M ] = annR M N , thus [N R : M ] is a semimaximal ideal. The following result follows immediately by lemma (2.7). 2.8 Proposi tion: If every submodule N of an R-module M is semimaximal, then M is coannsemimaximal. ..Next, we have the following remark. 2.9 Remark: The direct sum of coannsemimaximal modules need not be coannsemimaximal. For examp le: Let M =Z 4Z3 be a Z-module. Z4 and Z3 are two coannsemimaximal Z-modules. But M � Z12 which is not coannsemimaximal. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Re ferences 1. Abdul-Razak, H.M ., (1999), Quasi-Prime M odules and Quasi-Prime Submodules, M .D. Thesis, Univ. of Baghdad. 2. Abdul-Al-Kalik, J.A., (2005), Primary M odules, M .D. T hesis, Univ. of Baghdad. 3. Abdul-Al-Kalik, J.A., (2009), On M ax-M odules, t o app ear. 4. Coodreal, K.R., (1976), Ring Theory-Non Singular Rings and M odules, M arcei-Dekker, New York and Basel. 5. Khalaf, Y.H., (2007), Semimaximal Submodules, Ph.D. Thesis, University of Baghdad. 6. 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Okay ama Univ., Vol.28, p p .119-131. 2011) 3( 24المجلد مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة وشبھ االعظمیة التالفة المضادة المقاسات شبھ االعظمیة التالفة ف، أنعام محمد علي هادي حاتم یحیى خل جامعة بغداد ،ابن الهیثم -كلیة التربیة،قسم الریاضیات 2010 أیلول 20:استلم البحث في 2011 شباط 8 : قبل البحث في الخالصة في . ن درسوا المقاسات التي تالف كل مقاس جزئي غیر صفري منها هو أولي، ابتدائي أو اعظميبعض الباحثی Mهذا البحث قدمنا ودرسنا المقاسات شبه االعظمیة التالفة والمقاسات شبه االعظمیة التالفة المضادة، حیث یدعى المقاس على التوالي ( Rعلى الحلقة Nاذا كان تالف )على التوالي شبه اعظمي تالف مضاد(شبه اعظمي تالف Rعلى الحلقة تالف M N .Mفي Nلكل مقاس جزئي غیر صفري Rهو مثالي شبه اعظمي في ) Rعلى الحلقة :الكلمات المفتاحیة Annsemimaximal module, semisimple module, semisimp le ring, semiprime module, max- module, unifor m module, Z-regu lar module, F-re gular module, Artinian module, flat module, coprime module, coannsemimaximal module.