a thesis submitted in partial fulfillment of the requirements for the degree of doctor of philosophy (mathematics) faculty of graduate studies mahidol university

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a thesis submitted in partial fulfillment of the requirements for the degree of doctor of philosophy (mathematics) faculty of graduate studies mahidol university

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PRIMENESS IN MODULE CATEGORY LE PHUONG THAO A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (MATHEMATICS) FACULTY OF GRADUATE STUDIES MAHIDOL UNIVERSITY 2010 COPYRIGHT OF MAHIDOL UNIVERSITY Thesis entitled PRIMENESS IN MODULE CATEGORY Ms Le Phuong Thao Candidate Lect Nguyen Van Sanh, Ph.D Major-advisor Asst Prof Chaiwat Maneesawarng, Ph.D Co-advisor Asst Prof Gumpon Sritanratana, Ph.D Co-advisor Prof Banchong Mahaisavariya, M.D., Dip Thai Board of Orthopedics Dean Faculty of Graduate Studies Mahidol University Prof Yongwimon Lenbury, Ph.D Program Director Doctor of Philosophy Program in Mathematics Faculty of Science Mahidol University Thesis entitled PRIMENESS IN MODULE CATEGORY was submitted to the Faculty of Graduate Studies, Mahidol University for the degree of Doctor of Philosophy (Mathematics) on 19 October, 2010 Ms Le Phuong Thao Candidate Prof Le Anh Vu, Ph.D Chair Lect Nguyen Van Sanh, Ph.D Member Asst Prof Gumpon Sritanratana, Ph.D Member Asst Prof Chaiwat Maneesawarng, Ph.D Member Prof Banchong Mahaisavariya, M.D., Dip Thai Board of Orthopedics Dean Faculty of Graduate Studies Mahidol University Prof Skorn Mongkolsuk, Ph.D Dean Faculty of Science Mahidol University iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude and appreciation to my major advisor, Dr Nguyen Van Sanh, for his constructive guidance, valuable advice and inspiring talks throughout my study period that has enabled me to carry out this thesis successfully I am greatly grateful for having the guidance and encouragement of my Co-Advisors, Asst Prof Dr Chaiwat Maneesawarng and Asst Prof Dr Gumpon Sritanratana I would also like to thank Prof Dr Dinh Van Huynh from the Center of Ring Theory, Ohio University, Athens, USA, and Prof Dr Le Anh Vu from Vietnam National University - Hochiminh City, Vietnam I would like to express my deep gratitude to Department of Mathematics, Mahidol University, for providing me with the necessary facilities and financial support Special thanks go to all the teachers and staffs of the Department of Mathematics for their kind help and support I would like to thank all of my friends in the research group for their help throughout my study period at Mahidol University I am very glad to express my thankful sentiment to Cantho University for the recommendation and encouragement My love and dedication offer wholly to my family, for their love, sincere, intention, encouragement and understanding support throughout my Ph D study at Mahidol University Le Phuong Thao Fac of Grad Studies, Mahidol Univ Thesis / iv PRIMENESS IN MODULE CATEGORY LE PHUONG THAO 5137143 SCMA/D Ph.D (MATHEMATICS) THESIS ADVISORY COMMITTEE: NGUYEN VAN SANH, Ph.D (MATHEMATICS), CHAIWAT MANEESAWARNG, Ph.D (MATHEMATICS), GUMPON SRITANRATANA, Ph.D (MATHEMATICS) ABSTRACT In modifying the structure of prime ideals and prime rings, many authors transfer these notions to modules There are many ways to generalize these notions and it is an effective way to study structures of modules However, from these notion definitions, we could not find any properties which are parallel to that of prime ideals In 2008, N V Sanh proposed a new definition of a prime submodule The definition was to let R be a ring, M a right R-module, and S be its endomorphism ring If any ideal I of S and any fully invariant submodule U of M, IU ⊂ X implies IM ⊂ X or U ⊂ X, then the fully invariant submodule X of M is called a prime submodule A fully invariant submodule is called semiprime if it equals an intersection of prime submodules With this new definition, we found many beautiful properties of prime submodules that are similar to prime ideals From Sanh’s definition of prime submodules, we constructed some new notions such as nilpotent submodules, nil submodules, a prime radical, a nil radical and a Levitzki radical of a right or left module M over an arbitrary associative ring R and described all properties of them as generalizations of nilpotent ideals, nil ideals, a prime radical, a nil radical and a Levitzki radical of rings In this research, we also transfered the Zariski topology of rings to modules KEY WORDS : PRIME SUBMODULES/ ZARISKI TOPOLOGY NILPOTENT SUBMODULES/ NIL SUBMODULES PRIME RADICAL/ NIL RADICAL/ LEVITZKI RADICAL 80 pages v CONTENTS Page ACKNOWLEDGEMENTS iii ABSTRACT iv CHAPTER I INTRODUCTION 1.1 On the primeness of modules and submodules 1.2 On problems of primeness of modules and submodules CHAPTER II BASIC KNOWLEDGE 2.1 Generators and cogenerators 2.2 Injectivity and projectivity 2.3 Noetherian and Artinian modules and rings 11 2.4 Primeness in module category 13 2.5 On Jacobson radical, prime radical, nil radical and Levitzki radical of rings CHAPTER III 24 A GENERALIZATION OF HOPKINS-LEVITZKI THEOREM 27 3.1 Prime submodules and semiprime submodules 27 3.2 Prime radical and nilpotent submodules 30 CHAPTER IV ON NIL RADICAL AND LEVITZKI RADICAL OF MODULES 38 4.1 Nil submodules 38 4.2 Nil radical of modules 41 4.3 Levitzki radical of modules 47 vi CONTENTS (cont.) Page CHAPTER V THE ZARISKI TOPOLOGY ON THE PRIME SPECTRUM OF A MODULE CHAPTER VI CONCLUSION 50 68 REFERENCES 71 BIOGRAPHY 80 Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / CHAPTER I INTRODUCTION Throughout the text, all rings are associative with identity and all modules are unitary right R-modules For special cases, we describe with a precision Let R be a ring and M be a right R-module Denote S = EndR (M ) for its endomorphism ring, Mod-R for the category of all right R-modules and R-homomorphisms 1.1 On the primeness of modules and submodules Prime submodules and prime modules have been appeared in many contexts Modifying the structure of prime ideals, many authors want to transfer this notion to right or left modules over an arbitrary associative ring By an adaptation of basic properties of prime ideals, some authors introduced the notion of prime submodules and prime modules and studied their structures However, these notions are valid in some cases of modules over a commutative ring such as multiplication modules, but for the case of non-commutative rings, nearly we could not find something similar to the structure of prime ideals In 1961, Andrunakievich and Dauns ([31], [71]) first introduced and investigated prime module Following that, a left R-module M is called prime if for every ideal I of R, and every element m ∈ M with Im = 0, implies that either m = or IM = In 1975, Beachy and Blair ([10], [11]) proposed another definition of primeness, for which a left R-module M is called a prime module if (0 :R M ) = (0 :R N ) for every nonzero submodule N of M This definition is used in the book [48] of Goodearl and Warfield in 1983, McConnel and Robson [77] in 1987 In 1978, Dauns ([4], [31], [71]) defined that a module M is a prime module if (0 :R M ) = A(M ), where A(M ) = {a ∈ R | aRm = 0, m ∈ M } For Le Phuong Thao Introduction / the class of submodules, he also created the definitions of prime submodules and semiprime submodules A submodule P of a left R-module M is called a prime submodule if for any element r ∈ R and any element m ∈ M such that rRm ⊂ P, then either m ∈ P or r ∈ (P :R M ), and a submodule N of M is called a semiprime submodule if N = M and for any elements r ∈ R and m ∈ M such that rn m ∈ N, then rm ∈ N Following Bican ([20]), we say that a left R-module M is B-prime if and only if M is cogenerated by each of its nonzero submodules It is easy to see that B-prime implies prime In [100], it is pointed out that M is B-prime if and only if L · HomR (M, N ) = for every pair L, N of nonzero submodules of M In 1983, Wisbauer ([19], [64], [100], [101]) introduced the category σ[M ], a the full subcategory of M od-R whose objects are M -generated modules Following him, a left R-module M is a strongly prime module if M is subgenerated by any of its nonzero submodules, i.e., for any nonzero submodule N of M, the module M belongs to σ[N ], or equivalently, for any x, y ∈ M, there exists a set of elements {a1 , · · · , an } ⊂ R such that annR {a1 x, · · · , an x} ⊂ annR {y} In 1984, Lu [72] defined that for a left R-module M and a submodule X of M , an element r ∈ R is called a prime to X if rm ∈ X implies m ∈ X In this case, X = {m ∈ M | rM ⊂ X} = (X : r) Then X is called a prime submodule of M if for any r ∈ R, the homothety hr : M/X → M/X defined by hr (m) = mr, where m ∈ M/X is either injective or zero This implies that (0 : M/X) is a prime ideal of R and the submodule X is called a prime submodule if for r ∈ R and m ∈ M with rm ∈ X implies either m ∈ X or r ∈ (X : M ) In 1993, McCasland and Smith ([4], [71], [74], [76]) gave a definition that a submodule P of a left R-module M is called a prime submodule if for any ideal I of R and any submodule X of M with IX ⊂ P, then either IM ⊂ P or X ⊂ P In 2002, Ameri [2] and Gaur, Maloo, Parkash ([42], [43]) examined the structure of prime submodules in multiplication modules over commutative rings Following them, a left R-module M is a multiplication module if every submodule X is of the form IM for some ideal I of R and M is called a weak multiplication Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / module if every prime submodule of M is of the form IM for some ideal I of R Although, multiplicative ideal theory of rings was first introduced by Dedekind and Noether in the 19th century, multiplication modules over commutative rings were newly created by Barnard [9] in 1980 to obtain a module structure which behaves like rings The structure of multiplication modules over noncommutative rings was first studied by Tuganbaev [97] in 2003 In 2004, Behboodi and Koohy [14] defined weakly prime submodules Following them, a submodule P of a module M is a weakly prime submodule if for any ideals I, J of R and any submodule X of M with IJX ⊂ P, then either IX ⊂ P or JX ⊂ P In 2008, Sanh ([86]) proposed a new definition of prime submodule Let R be a ring and M, a right R-module with its endomorphism ring S A fully invariant submodule X of M is called a prime submodule if for any ideal I of S and any fully invariant submodule U of M, I(U ) ⊂ X implies I(M ) ⊂ X or U ⊂ X A fully invariant submodule is called semiprime if it equals an intersection of prime submodules A right R-module M is called a semiprime module if is a semiprime submodule of M Consequently, the ring R is semiprime ring if RR is a semiprime module By symmetry, the ring R is a semiprime ring if R R is a semiprime left R-module In 2008, Sanh ([87]) studied the concepts of M -annihilators and of Goldie modules to generalize the concept of Goldie rings Following that definition, a right R-module M is called a Goldie module if M has finite Goldie dimension and satisfies the ascending chain condition for M -annihilators A ring R is a right Goldie ring if RR is Goldie as a right R-module It is equivalent to say that a ring R is a right Goldie ring if it has finite right Goldie dimension and satisfies the ascending chain condition for right annihilators By using some properties of prime modules and Goldie modules, we study the class of prime Goldie modules Le Phuong Thao The Zariski topology on the prime spectrum of a module / 66 that IN ⊂ P Then P M is a prime submodule of M and P M ⊃ IN (M ) = N Hence P M ⊃ r(N ) Since Ir(N ) = Hom(M, r(N )) ⊂ Hom(M, P M ) = P, we have √ √ Ir(N ) ⊂ IN It follows that r(N ) ⊂ IN (M ) Proposition 5.38 Let M be a right R-module which is a self-generator Let Y be a subset of X M If Y is irreducible, then J(Y ) is a prime submodule of M Proof By Theorem 5.5, we have V (N ) = V (N ) for any fully invariant submodule of M It is clear that J(Y ) is a proper fully invariant submodule of M Clearly, Y ⊂ V (J(Y )) Let I be an ideal of S and U be a fully invariant submodule of M such that IU ⊂ J(Y ) Then Y ⊂ V (J(Y )) ⊂ V (IU ) ⊂ V (IM ) ∪ V (U ) Since V (IM ), V (U ) are closed sets of X M and Y is irreducible, we have Y ⊂ V (IM ) or Y ⊂ V (U ) If Y ⊂ V (IM ) then P ⊃ IM, for all P ∈ Y It would imply that IM ⊂ J(Y ) If Y ⊂ V (U ) Then P ⊃ U, for all P ∈ Y It follows that U ⊂ J(Y ) Thus IM ⊂ J(Y ) or U ⊂ J(Y ), proving that J(Y ) is a prime submodule of M Corollary 5.39 Let M be a right R-module which is a self-generator Let N be a fully invariant submodule of M Then V (N ) is irreducible if and only if r(N ) is prime in M Consequently, X M is irreducible if and only if P(M) is a prime submodule of M Proof (⇒) by Proposition 5.38 (⇐) by Theorem 5.19 Proposition 5.40 Let M be a prime module Then Spec(M ) is a T1 space if and only if is the only prime submodule of M Proof (⇒) Since Spec(M ) is a T1 space, we have p dim M ≤ (by Theorem 5.23) Since Spec(M ) = ∅, we have dim M = 0, and hence is the maximal prime submodule of M Therefore, is the only prime submodule of M (⇐) Since is the only prime submodule of M, it is the only maximal prime submodule of M Thus p dim M = 0, and so Spec(M ) is a T1 space Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 67 Lemma 5.41 Let M be a right R-module Let Y = {P1 , , Pn } be a finite subset of X M Then cl(Y ) = n V (Pi ) i=1 n Pi ) We have V (Pi ) ⊂ Proof By Proposition 5.12, cl(Y ) = V (J(Y )) = V ( n V( i=1 n i=1 i=1 n Q ∈ V( Pi ) Then I i=1 n Pi n V (Pi ) ⊂ V ( Pi ), for all i = 1, , n; and hence Pi ) Conversely, let i=1 ⊂ IQ It follows that IP1 IPn ⊂ IQ Since Q is a i=1 prime submodule of M, we get IQ is a prime ideal of S Thus IPk ⊂ IQ , for some n k = 1, , n Therefore Q ∈ V (Pk ), and hence Q ∈ n n Pi ) ⊂ V( i=1 V (Pi ) This shows that i=1 V (Pi ) i=1 Proposition 5.42 Let M be a right R-module Let Y be a finite irreducible closed subset of X M Then Y = V (P ) for some P ∈ X M Proof Suppose that Y = {P1 , , Pn } is an irreducible closed subset of X M Then n Y = cl(Y ) = V (Pi ), by Lemma 5.41 Since Y is irreducible, we get Y = V (Pi ) i=1 for some i = 1, , n Le Phuong Thao Conclusion / 68 CHAPTER VI CONCLUSION Several authors have extended the notion of prime ideals to modules There are many way to generalize these notions, however, from these definitions, we could not find some properties parallel to that of prime ideals Recently, we are successful in introducing a new definition of such a kind of submodules and we roughly call them prime submodules (see [86]) Following this new definition, we proved many beautiful properties of prime submodules that are similar to that of prime ideals Theorem 3.2.11 in Chapter III can be considered as a generalization of Hopkins-Levitzki Theorem Theorem 3.2.11 Let M be a quasi-projective, finitely generated right R-module which is a self-generator If M is Artinian, then M is Noetherian, J(M ) is nilpotent and J(M ) = P (M ) Theorem 4.3.7 in Chapter IV gives us the relation between prime radical, Levitzki-radical, nil radical and Jacobson radical of a quasi-projective, finitely generated right R-module which is a self-generator Theorem 4.3.7 Let M be a quasi-projective, finitely generated right R-module which is a self-generator Then P (M ) ⊂ L(M ) ⊂ N (M ) ⊂ J(M ) There are many properties of the Zariski topology on the prime spectrum Spec(M ) are given in Chapter V Some of the main results are: Theorem 5.10 Let M be a right R-module Then the set B = {XfM | f ∈ S} forms a basis for the Zariski topology on X M Theorem 5.11 Let M be a right R-module If the natural map ψ of X M is surjective, then X M is compact Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 69 Theorem 5.15 Let M be a right R-module with the Zariski topology on Spec(M ) Then the following conditions are equivalent: (1) X M is a T0 space; (2) The natural map ψ : X M → X S is injective; (3) For P, Q ∈ X M , if V (P ) = V (Q), then P = Q; (4) |SpecP (M )| ≤ for any P ∈ Spec(S) Theorem 5.18 Let M be a right R-module with surjective natural map ψ : X M → X S Then the following conditions are equivalent: (1) X M is connected; (2) X S is connected Theorem 5.23 Let M be a right R-module Then X M is a T1 space if and only if dim M ≤ Theorem 5.28 Let M be a right R-module with |Spec(M )| ≥ If Spec(M ) is a Hausdorff space, then dim M = Recall that a nonzero right R-module M is called retractable if Hom(M, N ) = for any nonzero submodule N of M It is clear that if M is a self-generator then M is retractable In most of the results in this thesis, we used the conditions M is quasi-projective, finitely generated and self-generator and the following question arises Do we still have these properties of prime submodules if we replace the condition ”self-generator” by ”retractable”? Let M be a right R-module The intersection of all prime submodules of M is called the prime radical of M and denoted by P (M ) The prime radical Le Phuong Thao Conclusion / 70 P (R) of a ring R is the intersection of all prime ideals of R It is the smallest ideal of R If R is commutative, then P (R) is precisely the set of all nilpotent elements of R In general, the prime radical of a ring R is exactly the set of all strongly nilpotent elements of R ([100], 2.13) We also want to find the description for the prime radical P (M ) of M We wish to answer the following question Can we describe the prime radical of a module M by term of elements as in the case of P (R)? Many authors introduce and generalize a generalization of the Zariski topology of rings to modules and call it the classical Zariski topology of M Then they investigate the interplay between the module-theoretic properties of M and the topological properties of Spec(M ) (see [15], [16], [96]) Modules whose classical Zariski topology is respectively T1 , Hausdorff or cofinite are studied, and several characterizations of such modules are given By working with Sanh’s definition, a natural question arises: What happen with the Zariski topology on Spec(M ) by using Sanh’s definition? Our research group are studying prime submodule by using Sanh’s definition and we wish to answer the above questions in the future Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 71 REFERENCES [1] Majid M Ali, Idempotent and nilpotent submodules of multiplication modules, Comm in Algebra 36 (2008), 4620–4642 [2] R Ameri, On the prime submodules of multiplication modules, Internat J Math Math Sci., 27 (2003), 1715–1724 [3] F W Anderson and K R Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol 13, Springer-Verlag, Berlin - Heidelberg New York, 1992 [4] S Annin, Associated and Attached Primes Over Noncommutative Rings, Ph D Thesis, University of California, Berkeley, 2002 [5] S Annin, Attached primes over noncommutative rings, J Pure Appl Algebra, 212 (2008), 510–521 [6] E P Armendariz, Rings with dcc on essential left ideals, Comm Algebra, 19 (1980), 1945–1957 [7] E Artin, C J Nesbitt and R M Thrall, Rings with Minimum Conditions, Ann Arbor, University of Michigan, 1944 [8] M F Atiyah, I G Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, Reading, Massachusetts, 1969 [9] A Barnard, Multiplication modules, J Algebra, 71(1) (1981), 174–178 [10] J A Beachy, A characterization of prime ideals, J Indian Math Soc., 37 (1973), 343–345 [11] J A Beachy and W D Blair, Finitely annihilated modules and orders in artinian rings, Comm Algebra, 6(1) (1978), 1–34 Le Phuong Thao References / 72 [12] J A Beachy, Introductory Lectures on Rings and Modues, London Math Soc Student Texts, No 47, Cambridge Univ Press, 1999 [13] J A Beachy, M-injective modules and prime prime M-ideals, Communications in Algebra, Vol 30 (2002), 4649–4676 [14] M Behboodi and H Koohy, Weakly prime modules, Vietnam J Math., 32(2) (2004), 185–199 [15] M Behboodi and M R Haddadi, Classical Zariski topology of modules and spectral space I, International Electronic Journal of Algebra, Vol 4(2008), 104–130 [16] M Behboodi and M R Haddadi, Classical Zariski topology of modules and spectral space II, International Electronic Journal of Algebra, Vol 4(2008), 131–148 [17] M Behboodi, A generalization of Baer’s lower nilradical for modules, Journal of Algebra and Its Applications, Vol 6, No (2007), 337–353 [18] M Behboodi and M J Noori, Zariski-like topology on the classical prime spectrum of a modules, Bulletin if the Inanian Mathematical Society, Vol.35, No.1 (2009), 255–271 [19] K I Beider and R Wisbauer, Strongly semiprime modules and rings, Comm Moscow Math Soc., Russian Math Surveys, 48(1) (1993), 163–200 [20] L Bican, P Jambor, T Kepka and P Nemec, Prime and coprime modules, Fundamenta Math., 57 (1980), 33–45 [21] N Bourbaki, Elements of the History of Mathematics, Springer-Verlag, 1994 ¨ [22] Fethi Callialp and Unsal Tekir, On the prime radical of a module over a noncommutative ring, Taiwanese J of Math Vol.8, No.2 (2004), 337–341 Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 73 ¨ [23] Fethi Callialp and Unsal Tekir, On finite union of prime submodules , Pakistan J of Applied Sciences 2(11) (2002), 1016-1017 [24] V Camilo and M F Yousif, Continuous rings with acc on annihilators, Canada Math Bull 34 (1991), 642–644 [25] A Cayley, On the theory of elimination, Cambridge and Dublin Math J III, 116-120, reprinted in Collected Mathematical Papers I, 370-374, Cambridge University Press, Cambridge, 1889 [26] A W Chatters and C R Hajarnavis, Rings with Chain Conditions, Pitman Advanced Publishing Program, 1980 [27] K C Chowdhury, Goldie modules, Indian J Pure Appl Math., 19(7) (1988), 641–652 [28] J Clark and D V Huynh, A note on self-injective perfect rings, Quartl J Math Oxford 45(2) (1994), 13–17 [29] I S Cohen, On the structure and ideal theory of complete local rings, Trans Amer Math Soc 59 (1946), 54–106 [30] I S Cohen and A Seidenberg, Prime ideal and integral dependence, Bull Amer Math Soc 52 (1946), 252–261 [31] J Dauns, Prime modules, J Reine Angew Math., 298 (1978), 156–181 [32] J Dauns, Primal modules, Comm Algebra, 25(8) (1997), 2409–2435 [33] P G L Dirichlet, R Dedekind, Lecture on Number Theory, Providence, RI: London Math Soc ; 1999 [34] N V Dung, D V Huynh, P F Smith and R Wisbauer, Extending Modules, Pitman, London, 1996 [35] H M Edwards, The genesis of ideal theory, Arch Hist Ex Sci., 23 (1980), 321–378 Le Phuong Thao References / 74 [36] D Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, 1995 [37] C Faith, Algebra I: Rings, Modules and Categories, Grundl Math Wiss., Band 190, Springer-Verlag, Berlin - Heidelberg - New York, 1973 [38] C Faith, Algebra II: Ring Theory, Grundl Math Wiss., Band 191, Springer-Verlag, Berlin - Heidelberg - New York, 1976 [39] C Faith, Rings with ascending chain condition on annihilators, Nagoya Math J., 27 (1996), 179–191 [40] A W Fuller, On direct representations of quasi-injective and quasi-projective modules, Arch Math., 20 (1996), 495–502 [41] B J Gardner and R Wiegandt, Radical Theory of Rings, Marcel Dekker, New York–Basel 2004 [42] A Gaur, A K Maloo and A Parkash, Prime submodules in multiplication modules, International Journal of Algebra, 1(8) (2007), 375–380 [43] A Gaur and A K Maloo, Minimal prime submodules, International Journal of Algebra, 2(20) (2008), 953–956 [44] A W Goldie, The structure of prime rings under ascending chain conditions, Proc London Math Soc., (1958), 589–608 [45] A W Goldie, Semiprime rings with maximum conditions, Proc London Math Soc., 10(3) (1960), 201–220 [46] K R Goodearl, Singular Torsion and the Splitting Properties, Mem Amer Math Soc., Vol 124, American Mathematical Society, Providence, Rhode Island, 1972 [47] K R Goodearl, Ring Theory: Nonsingular Rings and Modules, Marcel Dekker Inc., New York and Basel, 1976 Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 75 [48] K R Goodearl and R B Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, Vol 61, Cambridge University Press, Cambridge, 2004 [49] Q D Hai, A G A Pedro and R L Sergio, On the Goldie dimension of rings and modules, J of Algebra, 305 (2006),937–948 [50] D Handelman and J Lawrence, Strongly prime rings, Trans Amer Math Soc., 211 (1975), 209–223 [51] D Handelman, Strongly semiprime rings, Pacific J Math., 60(1) (1975), 115–122 [52] P J Hilton and U Stambach, A course in Homological Algebra, Springer– Verlag, New York, 1971 [53] C Hopkins, Rings with minimal condition for left ideals, Ann Math 40 (1939), 712–730 [54] Thomas W Hungerford, Algebra, Springer–Verlag, New York, 1974 [55] D V Huynh and P Dan, On serial Noetherian rings, Arch Math 56 (1991), 552–558 [56] D V Huynh, On some symmetry questions for prime and non-prime rings, to appear [57] N Jacobson, The theory of rings, Amer Math Soc Surveys, Vol 2, American Mathematical Society, Providence, Rhode Island, 1943 [58] S K Jain, T Y Lam and A Leroy, On uniform dimensions of ideals in right nonsingular rings, J of Pure and Applied Algebra, 133: 117–139, 1988 [59] J P Jan, Projective injective modules, Pacific J Math (1959), 1103–1108 [60] James Jenkins and Patrick F Smith, On the prime radical of a module over a commutative ring, Comm in Algebra, 20(12) (1992), 3593–3602 Le Phuong Thao References / 76 [61] R E Johnson, Prime Rings, Duke Math J., 18 (1951), 799–809 [62] S Karimzadeh and R Nekooei, On demension of modules, Turk J Math., 31 (2007), 95–109 [63] F Kasch, Modules and Rings, London Mathematical Society Monograph, No 17, Academic Press, London - New York - Paris, 1982 [64] A Kaucikas and R Wisbauer, On strongly prime rings and ideals, Comm Algebra, 28(11) (2000), 5461–5473 [65] I Kleiner, From numbers to rings: the early history of ring theory, Elem Math., 53 (1998), 18–35 [66] A Koehler, Quasi-projective and quasi-injective modules, Pacific J Math., 36(3) (1971), 713–720 [67] T Y Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol 131, Springer-Verlag, Berlin - Heidelberg - New York, 1991 [68] T Y Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol 189, Springer-Verlag, Berlin - Heidelberg - New York, 1999 [69] T Y Lam, Exercises in classical Ring Theory, Problem Books in Mathematics, Springer - Verlag, Berlin - Heidelberg - New York, 1995 [70] Serge Lang, Algebra, Springer–Verlag, New York, 2002 [71] Ch Lomp and A J Pe˜ na P., A note on prime modules, Divul Math., 8(1) (2000), 31–42 [72] C P Lu, Prime submodules of modules, Comm Math Univ Sancti Pauli, 33(1) (1984), 61–69 [73] C P Lu, Spectra of modules, Communication in Algebra, 23(10) (1995), 3741–3752 Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 77 [74] R L McCasland and M E Moore, Prime submodules, Comm Algebra, 20(6) (1992), 1803–1817 [75] R L McCasland and M E Moore, On radical of submodules, Communication in Algebra, 19(5) (1991), 1327–1341 [76] R L McCasland and P F Smith, Prime submodules of noetherian modules, Rocky Mountain J Math., 23(3) (1993), 1041–1062 [77] J C McConnell and J C Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol 30, Amer Math Soc., Providence, Rhode Island, 2001 [78] S H Mohamed and B J Muller, Continuous and discrete modules, London Mathematical Society Lecture Note Series 147, Cambridge University Press, Cambridge, 1990 [79] M E Moore and S J Smith, Prime and radical submodules of modules over commutative rings, Communication in Algebra, Vol 30, No 10, (2002), 5037–5064 [80] M Nagata, On the theory of radicals in a ring, J Math Soc Japan, (1951), 330–344 [81] W K Nichoson and M F Yousif, Quasi-Frobenius rings, Cambridge University Press, 2003 [82] B L Osofsky, Rings all of whose finitely generated modules are injective, Pacific J Math., 14 (1982), 645–650 [83] K H Parshall, H M Wedderburn and the structure theory of algebras, Arch Hist Ex Sci., 32 (1985), 223–349 [84] Donald S Passman, "A Course In Ring Theory", AMS Chelsea Publishing, Amer Math Society-Providence, Rhode Island, 2004 Le Phuong Thao References / 78 [85] Joseph J Rotman, A first course in Abstract Algebra, Prentice Hall, Upper Saddle River, New Jersey 07458 [86] N V Sanh, N A Vu, K F U Ahmed, S Asawasamrit and L P Thao, Primeness in module category, Asian-European Journal of Mathematics, (1) (2010), 145–154 [87] N V Sanh, S Asawasamrit, K F U Ahmed and L P Thao, On prime and semiprime Goldie modules, Asian-European Journal of Mathematics, accepted for publication [88] N V Sanh, K F U Ahmed and L P Thao, On semiprime modules and chain conditions, Southeast Asian Bulletin of Mathematics, accepted for publication [89] N V Sanh and L P Thao, A generalization of Hopkins-Levitzki Theorem, Southeast Asian Bulletin of Mathematics, accepted for publication [90] N V Sanh and L P Thao, On nil radical and Levitzki radical of modules, to appear [91] N V Sanh and L P Thao, The Zariski topology on the prime spectrum of a module, to appear [92] B Satyanarayana, K S Prasad and D Nagaraju, A theorem on modules with finite Goldie dimension, Source: Soochows J of Math., 32 (2), 311–315 [93] Robert C Shock, A note on prime radical, J Math Soc Japan, Vol.24, No.3, 1972 [94] P F Smith, Radical submodules and Uniform dimension of modules, Turk J Math., 28 (2004), 255–270 [95] B Stenstr¨om, Rings of Quotients, Springer-Verlag, Berlin - Heidelberg - New York, 1975 Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 79 ¨ [96] Unsal Tekir, The Zariski Topology on the Prime Spectrum of a Module over Noncommutative Rings, Algebra Colloquium 16:4 (2009) 691-698 [97] A A Tuganbaev, Multiplication modules and ideals, J Math Sci New York, 136(4) (2006), 4110–4130 [98] B L Van der Waerden, A History of Algebra, Springer-Verlag, 1985 [99] J H M Wedderburn, On hypercomplex numbers, Proc London Math Soc., (1907), 77–118 [100] R Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Tokyo, 1991 [101] R Wisbauer, On prime modules and rings, Comm Algebra, 11(20) (1983), 2249–2265 Le Phuong Thao Biography / 80 BIOGRAPHY NAME Ms Le Phuong Thao DATE OF BIRTH 20 January, 1973 PLACE OF BIRTH Hau Giang, Vietnam INSTITUTIONS ATTENDED Cantho University, Vietnam, 1990-1994 Bachelor of Science (Math Edu.) Amsterdam University, 1998-2000 Master of Science (Pure Math.) Mahidol University, 2008-2011 Doctor of Philosophy (Mathematics) POSITION Lecturer, 1994–Present Dept of Mathematics, Fac of Education, Cantho University, Vietnam HOME ADDRESS 464 Cach Mang Thang Tam Street Cantho City, Vietnam E-MAIL lpthao@ctu.edu.vn [...]... ideal of S From these new definitions, the authors also introduced prime radical, nil radical and Levitzki radical of a right R-module M and investigated their properties in Chapter III and Chapter IV Another question is: Can we construct and generalize of the Zariski topology of rings to modules by using Sanh’s definition? The answer is positive in Chapter V of the thesis For the structure of the thesis, ... properties of nilpotent submodules of a module There are also given important results of prime radical of module Chapter IV provides the definition of nil submodule, nil radical and Levitzki radical of a module The relation of prime radical, nil radical and Levitzki radical of a module are also given in chapter IV The generalization of the Zariski topology of rings to modules is given in chapter V Finally,... submodules has a minimal element (2) A ring R is called right Noetherian (resp right Artinian) if the module RR is Noetherian (resp Artinian) (3) A chain of submodules of MR · · · ⊂> Ai−1 ⊂> Ai ⊂> Ai+1 ⊂> · · · (finite or infinite) is called stationary if it contains a finite number of distinct Ai Remarks (a) Clearly, the definitions above are preserved by isomorphisms (b) Noetherian modules are called modules... homomorphism (from a submodule of a module B) into an injective module A can be ”completed” to a ”full” homomor- Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 7 phism (from all of B) into A Injective module first appeared in the context of abelian groups The general notion for modules was first investigated by Baer in 1940 The theory of these modules was studied long before the dual notion of projective... guarantees that every Le Phuong Thao Basic knowledge / 14 nonzero ring has at least one prime ideal Definition 2.4.4 A prime ideal P in a ring R is called a minimal prime ideal if it does not properly contain any other prime ideals For instance, if R is a prime ring, then 0 is the unique minimal prime ideal of R Proposition 2.4.5 ([48], Proposition 3.3) Any prime ideal P in a ring R contains a minimal... two-sided ideals, then the prime radical of R is a nilpotent ideal Proposition 2.5.6 ([48], Corollary 4.14) For a right or a left Artinian ring R, the Jacobson radical is coincided with the prime radical Theorem 2.5.7 ([48], Theorem 3.11) Let R be a right or left Noetherian ring and let P be the prime radical of R Then P is a nilpotent ideal of R containing all the nilpotent right or left ideals of R Theorem... ideal Theorem 2.4.6 ([48], Theorem 3.4) In a right or left Noetherian ring R, there exist only finitely many minimal prime ideals, and there is a finite product of minimal prime ideals (repetitions allowed) that equals zero Definition 2.4.7 An ideal P in a ring R is called a semiprime ideal if it is an intersection of prime ideals (By convention, the intersection of the empty family of prime ideals of. .. right Artinian), then every finitely generated right R-module MR is Noetherian (resp Artinian) Fac of Grad Studies, Mahidol Univ Ph.D (Mathematics) / 13 (3) Every factor ring of right Noetherian (resp Artinian) ring is again right Noetherian (resp Artinian) 2.4 Primeness in module category In this section, before stating our new results we would like to list some basic properties from [48] Definition... Hopkins-Levitzki Theorem said that every right Artinian ring is right Noetherian It is well-known that not every Artinian ring is Noetherian, for example the Pr¨ ufer group Zp∞ is Artinian but not Noetherian In this chapter, we prove that if M is a Artinian quasi-projective finitely generated right R-module which is a self-generator, then it is Noetherian This result can be considered as a generalization of Hopkins-Levitzki... (II)(3) in Theorem 2.3.2 is called ascending chain condition, briefly ACC Thus, Theorem 2.3.2 asserts that a module M is Noetherian if it satisfies ACC, and Artinian if it satisfies DCC Corollary 2.3.3 ([63], Corollary 6.1.3) (1) If M is a finite sum of Noetherian submodules, then it is Noetherian; if M is a finite sum of Artinian submodules, then it is Artinian (2) If the ring R is right Noetherian (resp

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