Ansari-Toroghy, H.

Pourmortazavi, Seyed Sajad

Miskolc Mathematical Notes

H. Ansari-Toroghy

seyed Sajad Pourmortazavi

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On the prime spectrum of modules

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 16 (2015), No. 2, pp. 1233–1242 DOI: 10.18514/MMN.2015.1102ON THE PRIME SPECTRUM OF MODULESH. ANSARI-TOROGHY AND S. S. POURMORTAZAVIReceived 18 January, 2014Abstract. Let R be a commutative ring and let M be an R-module. Let us denote the set of allprime submodules ofM by Spec.M/. In this article, we explore more properties of strongly topmodules and investigate some conditions under which Spec.M/ is a spectral space.2010 Mathematics Subject Classification: 13C13; 13C99Keywords: prime spectrum, Zariski topology, spectral space1. INTRODUCTION, ETCThroughout this article, all rings are commutative with identity elements, and allmodules are unital left modules. N, Z, and Q will denote respectively the naturalnumbers, the ring of integers and the field of quotients of Z. If N is a subset of anR-module M , then N M denotes N is an submodule of M .LetM be an R-module. For any submodule N ofM , we denote the annihilator ofM=N by .N WM/, i.e. .N WM/D fr 2RjrM N g. A submodule P ofM is calledprime if P ¤M and whenever r 2R and e 2M satisfy re 2 P , then r 2 .P WM/ ore 2 P .The set of all prime submodule of M is denoted by Spec.M/ (or X ). For anyideal I ofR containing Ann.M/, I andR will denote I=Ann.M/ andR=Ann.M/,respectively. Also the map  W Spec.M/! Spec.R/ given by P 7! .P WM/ iscalled the natural map ofX . M is called primeful (resp. X -injective) if eitherM D 0or M ¤ 0 and the natural map  is surjective (resp. if either X D¿ or X ¤¿ andnatural map  is injective). (See [3, 11] and [13].)The Zariski topology on X is the topology  described by taking the set ˝ DfV.N /jN is a submodule of M g as the set of closed sets of X , where V.N /D fP 2X j.P WM/ .N WM/g [11].The quasi-Zariski topology on X is described as follows: put V .N / D fP 2X jP  N g and ˝ D fV .N /jN is a submodule of M g. Then there exists a topo-logy  on X having ˝ as the set of it’s closed subsets if and only if ˝ is closedunder the finite union. When this is the case,  is called a quasi-Zariski topology onX and M is called a top R-module [15].c 2015 Miskolc University Press1234 H. ANSARI-TOROGHY AND S. S. POURMORTAZAVILet Y be a topological space. Y is irreducible if Y ¤¿ and for every decomposi-tion Y DA1[A2 with closed subsets Ai  Y; i D 1;2, we have A1 D Y or A2 D Y .A subset T of Y is irreducible if T is irreducible as a space with the relative topology.For this to be so, it is necessary and sufficient that, for every pair of sets F , G whichare closed in Y and satisfy T F [G, T F or T G. Let F be a closed subset ofY . An element y 2 Y is called a generic point of Y if Y D cl.fyg/ (here for a subsetZ of Y , cl.Z/ denotes the topological closure of Z).A topological space X is a spectral space if X is homeomorphic to Spec.S/with the Zariski topology for some ring S . This concept plays an important rolein studying of algebraic properties of an R-module M when we a have a relatedtopology. For an example, when Spec.M/ is homeomorphic to Spec.S/, whereS is a commutative ring, we can transfer some of known topological properties ofSpec.S/ to Spec.M/ and then by using these properties explore some of algebraicproperties of M .Spectral spaces have been characterized by M. Hochster as quasi-compact T0-spaces X having a quasi-compact open base closed under finite intersection and eachirreducible closed subset of X has a generic point [9, p. 52, Proposition 4].The concept of strongly top modules was introduced in [2] and some of its prop-erties have been studied. In this article, we get more information about this class ofmodules and explore some conditions under which Spec.M/ is a spectral space forits Zariski or quasi-Zariski topology.In the rest of this article, X will denote Spec.M/. Also the set of all maximalsubmodules of M is denoted by Max.M/.2. MAIN RESULTSDefinition 1 (Definition 3.1 in [1]). LetM be anR-module. M is called a stronglytop module if for every submodule N of M there exists an ideal I of R such thatV .N /D V .IM/.Definition 2 (Definition 3.1 in [2]). LetM be anR-module. M is called a stronglytop module if M is a top module and  D  .Remark 1. Definition 1 and Definition 2 are equivalent. This follows from the factthat if N is a submodule of M , then by [11, Result 3], we haveV.N /D V..N WM/M/D V ..N WM/M/:Remark 2 (Theorem 6.1 in [11]). Let M be an R-module. Then the following areequivalent:(a) .X;/ is a T0 space;(b) The natural map of X is injective;(c) V.P / D V.Q/, that is, .P W M/ D .Q W M/ implies that P D Q for anyP;Q 2X ;ON THE PRIME SPECTRUM OF MODULES 1235(d) jSpecp.M/j  1 for every p 2 Spec.R/.Remark 3. (a) Let M be an R-module and p 2 Spec.R/. The saturation of asubmodule N with respect to p is the contraction of Np in M and denotedby Sp.N /. It is known thatSp.N /DN ec D fx 2M j tx 2N for some t 2R npg:(b) Let M be an R-module and N M . The radical of N , denoted by rad.N /,is the intersection of all prime submodules of M containing N ; that is,rad.N /DTP2V .N/P ([14]).(c) A topological space X is Noetherian provided that the open (respectively,closed) subsets of X satisfy the ascending (respectively, descending) chaincondition ([4, p. 79, Exercises 5-12]).Proposition 1. Let M be an strongly top module and  be the natural map of X .Then(a) .X;/D .X;/Š Im .(b) If X is Noetherian, then X is a spectral space.Proof. (a) By [15, Theorem 3.5] and Remark 2,  jIm is bijective. Also we have .V.N //D f.P WM/jP 2X;.P WM/ .N WM/g:Now by [11, Proposition 3.1] and the above arguments,  is continuous and a closedmap. Consequently we have .X;/D .X;/Š Im .(b) Let Y D V .N / be an irreducible closed subset of X . Now by [6, Theorem3.4], we haveV .N /D V .rad.N //D cl.frad.N /g/:Hence Y has a generic point. Also X is Noetherian and it is a T0-space by [6, Pro-position 3.8 (i)]. Hence it is a spectral space by [9, Pages 57 and 58]. An R- module M is said to be a weak multiplication module if either X DSpec.M/D¿ orX ¤¿ and for every prime submodule P ofM , we have P D IMfor some ideal I of R (see [5]).The following theorem extends [1, Proposition 3.5], [1, Corollary 3.6], [1, The-orem 3.9 (1)], and [1, Theorem 3.9 (7)]. In fact, in part (a) of this theorem, wewithdraw the restrictions of finiteness and Noetherian property from [1, Proposition3.5] and [1, Corollary 3.6], respectively. In part (b), we remove the conditions “ Mis primeful ” and “ R is a Noetherian ring ” in [1, Theorem 3.9 (1)] and instead ofthem, we put the weaker conditions “ Im. / is closed in Spec.R/ ” and “ Spec.R/is a Noetherian space ”. In part (c), we withdraw the condition “ R has Noetherianspectrum ” from [1, Theorem 3.9 (7)] and put the weaker condition “ the intersectionof every infinite family of maximal ideals of R is zero ”.Theorem 1. Let M be an R-module. Then we have the following.1236 H. ANSARI-TOROGHY AND S. S. POURMORTAZAVI(a) Let .Mi /i2I be a family of R-modules and let M D ˚i2IMi . If M is anstrongly top R-module, then each Mi is an strongly top R-module.(b) If M be an strongly top R-module and  be the natural map of X , then wehave(i) If Im. / is closed in Spec.R/, then .X;/ D .X;/ is a spectralspace.(ii) If Spec.R/ is Noetherian, then .X;/D .X;/ is a spectral space.(c) Suppose R is a one dimensional integral domain such that the intersection ofevery infinite family of its maximal ideals is zero. If M is a weak multiplica-tion R-module, then M is a top module.Proof. (a) Each Mi is a homomorphic image of M , hence it is strongly topby [1, Proposition 3.3].(b) (i) By Proposition 1, we have .X;/D .X;/Š Im. /. Now the claimfollows by [11, Theorem 6.7].(ii) As Spec.R/ is Noetherian, Im. / is also Noetherian. Now the claimfollows from Proposition 1.(c) Use the technique of [3, Theorem 3.18].The following theorem extends [1, Theorem 3.9(3) ].Theorem 2. Suppose R is a one dimensional integral domain such that the inter-section of every infinite family of its maximal ideals is zero. If M is X -injective withS0(0) rad (0), then M is a top module.Proof. If S0.0/ DM , then X D ¿ and there is nothing to prove. Otherwise, by[12, Corollary 3.7], S0.0/ is a prime submodule so that S0.0/D rad.0/. Hence thenatural map f W Spec.M=S0.0//! Spec.M/ is a homeomorphism by [7, Proposi-tion 1.4]. But by [3, Theorem 3.7 (a)] and [3, Theorem 3.15 (e)], M=S0.0/ is a weakmultiplication module. Now the result follows because by Theorem 1 (c), M=S0.0/is a top module. Let M be an R-module. Then M is called a content module if for every x 2M ,x 2 c.x/M , where c.x/ DTfI jI is an ideal of R such that x 2 IM g (see [13, p.140]).In below we generalize [1, Theorem 3.9(4)].Theorem 3. Suppose R is a one dimensional integral domain and let M be acontent R-module. Then we have the following.(a) If M is X -injective , then M is a top module.(b) If M is X -injective and S0(0)  rad (0), then M is an strongly top module.Furthermore, if Spec.R/ is Noetherian, then .X;/ is spectral.ON THE PRIME SPECTRUM OF MODULES 1237Proof. (a) By [3, Theorem 3.21], we haveSpec.M/D fSp.pM/jp 2 V.Ann.M//;Sp.pM/¤M g D fS0.0/g[Max.M/;whereMax.M/D fpM jp 2Max.R/;pM ¤M g:Let N M and let N ª S0.0/. Thenrad.N /D\NP2Spec.M/P D\NP2Max.M/P:So by the above arguments, there is an index set I such that rad.N /DTi2I .piM/.Since M is content module,V .N /D V .rad.N //D V .\i2I.piM//D V..\i2IpiM/:Now if N  S0.0/, then by [10, Lemma 2],V .N /D V .rad.N //D V .S0.0/\ .\i2I.piM///D V .S0.0/\ ..\i2Ipi /M//D V .S0.0//[V ..\i2Ipi /M/D V .S0.0//[V..\i2Ipi /M/:By the above arguments, it follows that M is a top module.(b) By [3, Theorem 3.21],Spec.M/D fS0.0/g[Max.M/ and Max.M/D fpM jp 2Max.R/;pM ¤M g:Let N  M . If N  S0.0/, then V .N / D V .0/ D X . Otherwise, we haverad.N /DTi2I .piM/ by [3, Theorem 3.21]. SinceM is content, by [11, Result 3]we haveV .N /D V .rad.N //D V .\i2I.piM//D V..\i2Ipi /M/:Hence M is an strongly top module. The second assertion follows from Theorem 1(b). Theorem 4. If M is content weak multiplication, then M is an strongly top mod-ule. Moreover, if Spec.R/ is Noetherian, then .X;/ is a spectral space.1238 H. ANSARI-TOROGHY AND S. S. POURMORTAZAVIProof. Let N M . Then we haveV .N /D V .rad.N //D V .\NPP /:Since M is a weak multiplication module, for each prime submodule P of M con-taining N , there exists an ideal IP of R such that P D IPM . Hence since M is acontent module,V .N /D V .\NP.IPM//D V ..\NPIP /M/:This implies that M is an strongly top module. Since Spec.R/ is Noetherian, so isSpec.R/. Hence by Theorem 1 (b), .X;/ is a spectral space. Theorem 5. Let R be a one-dimensional integral domain and let M be an X -injective R-module such that S0(0)  rad (0). If the intersection of every infinitenumber of maximal submodules of M is zero, then M is strongly top and .X;/ isa spectral space.Proof. If S0.0/ D M , then X D ¿ and there is nothing to prove. Otherwise,by [3, Theorem 3.21], we have Spec.M/ D fS0.0/g[Max.M/ and Max.M/ DfpM jp 2Max.R/;pM ¤M g. Now letN M . IfN D 0, then claim clear becauseV .N /D V .0/D V .0M/D Spec.M/. So we assume that N ¤ 0. We considertwo cases.(1) N  S0.0/. In this case, we have V .N /D V .0/D V .0M/D Spec.M/.(2) N ª S0.0/. Then since N ¤ 0 and the intersection of every infinite num-ber of maximal submodules of M is zero, rad.N / DTniD1.piM/, where piM 2Max.M/ for each i .1 i  n/. Hence we haveV .N /D V .rad.N //D V .n\iD1.piM//:Now we show that V .TniD1.piM// D V ..TniD1pi /M/. Clearly,V .TniD1.piM//  V ..TniD1pi /M/. Too see this reverse inclusion, letP 2 V ..TniD1pi /M/. If P D S0.0/, then .TniD1pi /M  S0.0/ implies thatTniD1pi  ..TniD1pi /M WM/  .S0.0/ WM/ D 0. Thus, there exists j .1  j n/ such that pj D 0, a contradiction. Hence we must have P D qM , where q 2Max.R/. Then, similar the above arguments, there exists j .1  j  n/ such thatq D pj . Therefore, P D qM D pjM 2 V .TniD1.piM//. So we haveV .N /D V .n\iD1.piM//D V ..n\iD1pi /M/:Hence M is strongly top so that  D . On the other hand,  D  is a subsetof a finite complement topology. This implies that .X;/ is Noetherian. Now byProposition 1, .X;/D .X;/ is spectral. ON THE PRIME SPECTRUM OF MODULES 1239Theorem 6. If for each submodule N of M , rad.N / Dp.N WM/M , then Mis an strongly top module. Moreover, if Spec.R/ is Noetherian , then .X;/ isspectral.Proof. Let N M . Then we haveV .N /D V .rad.N //D V .p.N WM/M/D V.p.N WM/M/D V.rad.N //D V.N /:Hence M is an strongly top module. Now the result follows by using similar argu-ments as in the proof of Theorem 4. Remark 4. Theorems 4, 5, and 6 improve respectively [1, Theorem 3.9(5)], [1,Theorem 3.9(8)], and [1, Theorem 3.9(6)]. They show that the notion of ” top mod-ules ” can be replaced by ” strongly top modules ” and the proofs can be shortenedconsiderably.In below we generalize [1, Theorem 3.36].Theorem 7. Let M be a primeful R-module. Then we have the following.(a) If .X;/ is discrete, then Spec.M/DMax.M/.(b) If R is Noetherian and Spec.M/DMax.M/, then .X;/ is a finite discretespace.Proof. (a) Since .X;/ is discrete, it is a T1-space. Now by [3, Theorem 4.3], wehave Spec.M/DMax.M/.(b) By [3, Theorem 4.3], Spec.R/DMax.R/. Hence R is Artinian. Now by [3,Theorem 4.3], .X;/ is a T0-space. Thus by Remark 2,M isX -injective. ButM is acyclic R-module and hence a cyclic R-module by [3, Remark 3.13] and [3, Theorem3.15]. Also .Spec.M/;/ is homoeomorphic to Spec.R/ by [11, Theorem 6.5(5)].Hence X is a finite discrete space by [4, Chapter 8, Exe 2].It is well known that if R is a PID and Max.R/ is not finite, then the intersectionevery infinite number of maximal ideals of R is zero. Now it is natural to ask thefollowing question: Is the same true when R is a one dimensional integral domainwith infinite maximal ideals? In below, we show that this true when Spec.R/ is aNoetherian space. Although this is not a simple fact, it used by some authors withoutgiving any proof.Theorem 8. (a) Let I be an ideal of R and let k;n 2N. Then .pI W ak/D.pI W an/.(b) Let I be an ideal of R and let a 2 R, n 2 N. Then pI Dq.pI W an/\qhpI ;ani.1240 H. ANSARI-TOROGHY AND S. S. POURMORTAZAVI(c) Suppose Spec.R/ is a Noetherian topological space. Then for every ideal Iof R,pI has a primary decomposition.(d) Suppose R is a one dimensional integral domain and Spec.R/ is a Noeth-erian topological space. Then the intersection of every infinite number ofmaximal ideals is zero.Proof. (a) It is clear.(b) Let f 2q.pI W an/\qhpI ;ani. Then there ism 2N such that f m 2 .pI Wan/\hpI ;ani. It follows that f m D gCxan for some g 2 pI and x 2 R and wealso get anf m 2 pI . Hence anf m D angCxa2n. This implies that xa2n 2 pIand so x 2 .pI W an/ by part (a). Thus xan 2 pI . It follows that f 2 pI . Thereverse inclusion is clear.(c) Set ˙ DfpI jI is a proper ideals of R andpI doesn’t have any primary decompositiong:Since Spec.R/ is Noetherian, the radicals of ideals satisfy the a.c.c. condition. So˙ has a maximal member,pI0 say. ThuspI0 … Spec.R/. In other words,9a;b 2R s.t. ab 2pI0 and a …pI0 and b …pI0:By part (b) we havepI0 Dp.pI0 W b/\phpI0;bi. Further,pI0 ¨p.pI0 W b/andpI0 ¨phpI0;bi. Sincep.pI0 W b/ andphpI0;bi have primary decomposi-tions by hypothesis,pI0 has a primary decomposition, a contradiction.(d) Since R is one dimensional integral domain, Spec.R/D f0g[Max.R/. Sup-pose fmigi2I is an infinite family of maximal ideals of R such thatTi2I mi ¤ 0. Bypart (c),pTi2I mi has a primary decomposition. Hences\i2Imi Dn\jD1m0j ; m0j 2Max.R/:This implies that fmigi2I is a finite family, a contradiction. So the proof is compe-leted. Example 1. We show thatZŒip5 is a one dimensional Noetherian integral domainwhich has infinite number of maximal ideals and it is not a PID. To see this, let WZŒX!ZŒip5 be the natural epimorphism given by p.x/ 7! p.ip5/. by using[8] or [16], one can see thatSpec.ZŒX/D fhpi;hf i;hq;gijp and q are prime numbers, f is a primaryirreducible polynomial in QŒX, and g is an irreducible polynomial in ZqŒXg:Now we have ker D hX2C5i. A simple verification shows thatSpec.ZŒip5/D f0g[Max.ZŒip5/ON THE PRIME SPECTRUM OF MODULES 1241D f0g[fhq;g.p 5/ij hq;gi 2 Spec.ZŒX/ and X2C5 2 hq;gig:Further ZŒip5 contains a finite number elements which are invertible by [17, p.38]. SoZŒip5 is a Noetherian one dimensional integral domain with infinite numberof maximal ideals. Hence the intersection of every infinite number of maximal idealsof ZŒip5 is zero by Theorem 8 (c). Note that ZŒip5 is not a PID by [17, p. 38].ACKNOWLEDGEMENTThe authors would like to thank the referee for the careful reading of our manu-script and valuable comments.REFERENCES[1] A. Abbasi and D. Hassanzadeh-Lelekaami, “Modules and spectral spaces,” Comm. Algebra,vol. 40, no. 11, pp. 4111–4129, 2012, doi: 10.1080/00927872.2011.602273.[2] H. Ansari-Toroghy and S. Keyvani, “Strongly top modules,” Bull. Malaysian. Math. Sci. Soc.(2),vol. 37, no. 1, pp. 73–82, 2014.[3] H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh, “On the prime spectrum of X-injective modules,”Comm. Algebra, vol. 38, no. 7, pp. 2606–2621, 2010, doi: 10.1080/00927870903036339.[4] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra. Addison-Wesley Read-ing, 1969, vol. 2.[5] A. Azizi, “Weak multiplication modules,” Czechoslovak Math. J., vol. 53, no. 3, pp. 529–534,2003, doi: 10.1023/B:CMAJ.0000024500.35257.39.[6] M. Behboodi and M. Haddadi, “Classical Zariski topology of modules and spectral spaces I,”International Electronic Journal of Algebra, vol. 4, pp. 104–130, 2008.[7] M. Behboodi and M. Haddadi, “Classical Zariski topology of modules and spectral spaces II,”International Electronic Journal of Algebra, vol. 4, pp. 131–148, 2008.[8] D. Eisenbud and J. Harris, The Geometry of Schemes. New York, Inc: Springer-Verlag, 2000.[9] M. Hochster, “Prime ideal structure in commutative rings,” Trans. Amer. Math. Soc., vol. 142, pp.43–60, 1969, doi: 10.1090/S0002-9947-1969-0251026-X.[10] C. P. Lu, “M-radicals of submodules in modules,” Math. Japon., vol. 34, no. 2, pp. 211–219, 1989.[11] C. P. Lu, “The Zariski topology on the prime spectrum of a module,” Houston J. Math., vol. 25,no. 3, pp. 417–432, 1999.[12] C. P. Lu, “Saturations of submodules,” Comm. Algebra, vol. 31, no. 6, pp. 2655–2673, 2003, doi:10.1081/AGB-120021886.[13] C. P. Lu, “A module whose prime spectrum has the surjective natural map,” Houston J. Math.,vol. 33, no. 1, pp. 125–143, 2007.[14] R. McCasland and M. Moore, “On radicals of submodules,” Comm. Algebra, vol. 19, no. 5, pp.1327–1341, 1991, doi: 10.1080/00927879108824205.[15] R. McCasland, M. Moore, and P. Smith, “On the spectrum of a module over a commutative ring,”Comm. Algebra, vol. 25, no. 1, pp. 79–103, 1997, doi: 10.1080/00927879708825840.[16] D. Mumford, The Red Book of Varieties and Schemes. Springer-Verlag, 1980.[17] D. Sharpe, Rings and factorization. CUP Archive, 1987.1242 H. ANSARI-TOROGHY AND S. S. POURMORTAZAVIAuthors’ addressesH. Ansari-ToroghyUniversity of Guilan, Department of pure Mathematics, Faculty of mathematical Sciences, P. O. Box41335-19141, Rasht, Iran.E-mail address: ansari@guilan.ac.irS. S. PourmortazaviUniversity of Guilan, Department of pure Mathematics, Faculty of mathematical Sciences, P. O. Box41335-19141, Rasht, Iran.E-mail address: mortazavi@phd.guilan.ac.ir

ON THE PRIME SPECTRUM OF MODULES

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