IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Annsemimaximal and Coannsemimaximal Modules I. M.A.Hadi, H. Y. Khalaf Department of Mathematics, College of Education Ibn-Al-Haitham University of Baghdad Received in : 20 September 2010 Accepted in : 8 February 2011 Abstract Some authors studied modules with annihilator of every nonzero submodule is p rime, primary or maximal. In this paper, we introduce and study annsemimaximal and coannsemimaximal modules, where an R-module M is called annsemimaximal (resp . coannsemimaximal) if annRN (resp . R M ann N ) is semimaximal ideal of R for each nonzero submodule N of M . Keywords: Annsemimaximal module, semisimple module, semisimple ring, semiprime module, max-module, uniform module, Z-regular module, F-regular module, Artinian module, flat module, coprime module, coannsemimaximal module. Introduction Let R be a commutative ring with unity and let M be an R-module. Muntaha A.R.H. in [1] introduced and studied quasi-prime modules where an R-module M is quasi-prime if annRN is a prime ideal of R for every nonzero submodule N of M. Adwia J.A.A. in [2] introduced and studied quasi-primary modules, where an R-module M is called quasi-primary if annRN is a primary ideal of R, for each nonzero submodule N of M. Adwia J.A.A. in [3] introduced and studied 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 paper, we introduced and studied 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- Annsemimaximal Modules In this section, we introduce the concept of annsemimaximal modules. We give some characterizations to this concept and establish some basic properties of this concept. 1.1 Definition: 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. SCI. 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 epimorphism 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,prp.(1.2.11)]. Thus   is annsemimaximal module. (9) The homomorphic image of annsemimaximal module is annsemimaximal module. Proof: Let f:MM' be an epimorphism such that M is annsemimaximal module. Then by the first fundamental theorem of homomorphisim, 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 Proposition: 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,prop. (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 proposition (1.3) M is annsemimaximal module  annRM is a semimaximal ideal.  R/annRM is semisimple ring. Now, we have the following theorem. 1.5 Theorem: 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. SCI. 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), suppose 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 prop. (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 semisimple R- module, [6]. 1.6 Proposition: Every semisimple R-module M is annsemimaximal. Proof: By [6,prop.(1.1.46)], we get R/annRM is a semisimple ring. Therefore annRM is a semimaximal ideal by [4,prop.(1.2.5)]. Thus M is annsemimaximal by prop. (1.3). The following corollary is an application of proposition (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,prop.(1.1.44)]. Hence M is annsemimaximal module by previous proposition. Next, we have the following proposition. 1.8 Proposition: 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 proposition (1.8). 1.10 Corollary: Let M is an Artinian R-module.Then M is semisimple module if and only if M is annsemimaximal. 1.11 Proposition: If M is annsemimaximal R-module, then every cyclic submodule of M is semisimple. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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,prop.(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 previous proposition. Therefore M= n i i 1 Rx   is semisimple By combining corollary (1.12), proposition (1.6), we get the 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 semisimple ring if and only if R is annsemimaximal ring. Now, we turn our attention to direct sum of annsemimaximal modules. 1.15 Proposition: 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 proposition (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), proposition (1.15) and corollary (1.14), we get the following: 1.16 Corollary: Let M be a faithful finitely generated R-module. The following statements are equivalent: (1) M is annsemimaximal. (2) M is semisimple. (3) R is semisimple. (4) R is annsemimaximal. Now, we give the following proposition. 1.17 Proposition: 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 Proposition: 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. Thus M 1 and M2 are annsemimaximal modules by remarks and examples ((1.2),(9)). () we have annRM 1 is semimaximal ideal and annRM 2 is semimaximal by proposition (1.3). On the 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. SCI. VOL.24 (3) 2011 annRM 1 annRM 2 is semimaximal. Therefore annR(M 1M 2) is semimaximal. Thus M 1M 2 is annsemimaximal module, by prop.(1.3). .. Recall that an R-module M is called semiprime 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 Proposition: Every annsemimaximal R-module is semiprime 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,prop.(1.2.21)], annRN is semiprime and hence M is a semiprime module. The converse of this proposition is not true in general. For example:Z as a Z-module is semiprime module, but it is not annsemimaximal module by remarks and examples ((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 respect to inclusion, [6], [7]. An R-module M is said to be a prime module if annRM=annRN for every non-zero submodule N of M , [8], [9]. As an application of proposition (1.19), we give the following corollary. 1.20 Corollary: Let M be a serial annsemimaximal module. Then M is prime R-module. Proof: From proposition (1.19), M is semiprime module and from [7,prop.(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 semiprime module are equivalent. 1.21 Proposition: Let M be a max-module. Then M is annsemimaximal module if and only if M is semiprime module. Proof: Suppose that M is semiprime 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 implies 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 proposition (1.19). Now, the following results are other consequences of proposition (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. SCI. 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 proposition (1.21) and [7,prop.(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 semiprime by proposition (1.21) and by [7,prop.(4.2.4)], we get the result. Now, we can give the following proposition. 1.24 Proposition: Let M be a uniform max-R-module. Then the following statements are equivalent. (1) M is annsemimaximal module. (2) M is seiprime module. (3) M is quasi-Dedekind. (4) M is prime. Proof: (1)  (2) by proposition (1.19). (2)  (3) by [7,prop.(4.2.4)]. (3)  (4) by [11,prop.(1.7), ch.2]. (4)  (1) It is clear that every prime module is semiprime module and hence by proposition (1.21) we get the 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,prop.(1.3.5)], M is regular module. 1.26 Proposition: 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,prop.(1.2.11)]. Hence M/N is regular R-module by [5,prop.(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 pure [12,ch.2]. By using this concept, we give the following proposition. 1.28 Proposition: If M is annsemimaximal R-module, then M is F-regular. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Proof: We have M is annsemimaximal, then annRM is semimaximal. Thus every cyclic submodule is pure by [5,prop.(1.3.9)]. Hence M is F-regular. 1.29 Proposition: Let R be a PID, annRM0, M is prime R-module. Then M is annsemimaximal module. Proof: Since M is prime R-module. Then annRM is prime ideal which implies that annRM is maximal ideal (since R is PID). Thus annRM is semimaximal ideal of R. Hence M is annsemimaximal R-module, by proposition (1.3). The converse of proposition (1.29) is not true, for example:Z6 as Z-module is annsemimaximal. But M is not prime. Recall that an R-module M is flat if for each injective homomorphisim 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 Proposition: If M is flat annsemimaximal R-module, then every homomorphic image of M is flat. Proof: We have M is annsemimaximal, then annRM is semimaximal ideal. Thus by [5,prop.(1.3.10)], we get the result. Next, we introduce the following definition. 1.31 Definition: Let N be a proper 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,prop.(1.2.11)], we get [N R : (m)] is semimaximal ideal of R. We end this section by the following result. 1.34 Proposition: 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: Suppose 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. SCI. VOL.24 (3) 2011 2- Coannsemimaximal Modules: In this section, we introduce the concept of coannsemimaximal module which is strongly from the concept of annsemimaximal module in section one. We give some characterizations about this concept and many results are studied. We start with the following definition. 2.1 Definition: An R-module M is called coannsemimaximal module if annR M N is semimaximal ideal of R for each non-zero proper submodule N of M. Equivalently, M is coannsemimaximal if M N is annsemimaximal module for each non-zero proper submodule N of M. 2.2 Examples: (1) Z12 is not coannsemimaximal Z-module, since if N = < 4 >, then annZ 12Z N = annZZ4 = 4Z which is not semimaximal ideal. (2) 2p Z as a Z-module is coannsemimaximal, where p is a prime number. Proof: Since < p > is only non-zero proper submodule of 2p Z . 2p Z /< p > � Zp and annZZp=pZ which is clear semimaximal ideal. Next, we have the following proposition. 2.3 Proposition: 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 examples ((1.2),(9)). Thus annR M N is semimaximal which implies that M is coannsemimaximal module. The converse of proposition (2.3) is not true in general. For example: 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 proposition proves 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 proper submodule N of M , [13]. 2.4 Proposition: 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 proper submodule N of M. But annR M N is semimaximal ideal of R for each non-zero proper submodule N of M (since M is coannsemimaximal). Thus annnRM is semimaximal ideal of R and hence M is annsemimaximal module by proposition (1.3). IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 As an application 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 proposition (2.4), we get the result. . Recall that a non-simple R-module M is called antihopfian if MM/N for all proper submodules N of M, [15]. By using this concept we get the following. 2.6 Proposition: Let M be an antihopfian, 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 prop.(1.5). But M M W N � for each proper submodule W of M, since M is antihopfian. That means M M N . Thus M W is annsemimaximal for all proper submodule W of M. Therefore M is coannsemimaximal module. Now, we prove 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: Suppose that N is a semimaximal submodule. Then by [5,def.(2.1.1)], M N is semisimple R-module and hence by proposition (1.6), M N is annsemimaximal module. Then by proposition (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 Proposition: 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 example: Let M=Z4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. SCI. VOL.24 (3) 2011 References 1. Abdul-Razak, H.M ., (1999), Quasi-Prime Modules and Quasi-Prime Submodules, M.D. Thesis, Univ. of Baghdad. 2. Abdul-Al-Kalik, J.A., (2005), Primary Modules, M .D. Thesis, Univ. of Baghdad. 3. Abdul-Al-Kalik, J.A., (2009), On M ax-Modules, to appear. 4. Coodreal, K.R., (1976), Ring Theory-Non Singular Rings and Modules, Marcei-Dekker, New York and Basel. 5. Khalaf, Y.H., (2007), Semimaximal Submodules, Ph.D. Thesis, University of Baghdad. 6. Kasch, F., (1982), M odules and Rings, Academic Press, London. 7. Al-Sharide, F.A.F., (2008), S-Compactly Packed Submodules and Semiprime Modules, Ms.C.Thesis, University of Tikrit. 8. Farzalipour, F. and Ghiasvand, P., (2009), Quasi-Multiplication Modules, Thai J. of Math., Vol.7(2), pp.361-366. 9. Desale, G. and Nicholson, W.K., (1981), Endoprimitive Rings, J.Algebra, Vol.70, pp.548-560. 10. 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Okayama Univ., Vol.28, pp.119-131. 2011) 3( 24المجلد مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة وشبھ االعظمیة التالفة المضادة المقاسات شبھ االعظمیة التالفة حاتم یحیى خلف، أنعام محمد علي هادي جامعة بغداد ،ابن الهیثم -كلیة التربیة،قسم الریاضیات یلول 20:استلم البحث في 2010 أ 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, semisimple ring, semiprime module, max- module, uniform module, Z-regular module, F-regular module, Artinian module, flat module, coprime module, coannsemimaximal module.