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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 1.1 On the primeness of modules and submodules . . . . . . . . . . . . 1 1.2 On problems of primeness of modules and submodules 4 CHAPTER II . . . . . . . BASIC KNOWLEDGE 5 2.1 Generators and cogenerators . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Injectivity and projectivity . . . . . . . . . . . . . . . . . . . . . . . 6 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) / 1 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 = 0 or IM = 0. 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 / 2 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 6= 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 ) 6= 0 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) / 3 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 0 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 1.2 Introduction / 4 On problems of primeness of modules and submodules Recently, Sanh ([89], [90]) introduced the notions of nilpotent submod- ules and nil submodules. Let M be a right R-module and X, a submodule of M. We denote IX = {f ∈ S | f (M ) ⊂ X}. We say that X is a nilpotent submodule of M if IX is a right nilpotent ideal of S. A submodule X of M is called a nil submodule of M if IX is a right nil 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, Chapter I is the introduction, Chapter II contains basic knowledge, and main results are included in Chapters III, IV and V. About the content of the study, Chapter I mentions preceding primeness concepts in the module category which generalized the primeness in ring theory. Chapter II provides essential basic knowledge that is needed for the study. Chapter III deals with the formal definition, basic 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, we review and conclude the results in Chapter VI. Fac. of Grad. Studies, Mahidol Univ. Ph.D. (Mathematics) / 5 CHAPTER II BASIC KNOWLEDGE Throughout this thesis, R is an arbitrary ring and Mod-R, the category of all unitary right R-modules. The notation MR indicates a right R-module M and S = EndR (M ) for its endomorphism ring. The set Hom(M, N ) denotes the set of right R-module homomorphisms between two right R-modules M and N and if further emphasis is needed, the notation HomR (M, N ) is used. A submodule X of M is indicated by writing X ⊂> M. Also I ⊂> RR means that I is a right ideal of R and I ⊂>R R that I is a left ideal. The notation I ⊂> R is reserved for two-sided ideals. The result in this chapter can be found in [3], [53], [63], [67], [68], [86], [87], [88], [95], [100]. 2.1 Generators and cogenerators Generators and cogenerators are notions in categories. They play an important role in Module Theory and in some categories. Below we will review these notions. Definition 2.1.1 (a) A module BR is called a generator for Mod-R, if P ∀M ∈ Mod-R[M = Imϕ]. ϕ∈HomR (B,M ) (a) A module CR is called a cogenerator for Mod-R, if T ∀M ∈ Mod-R[0 = Kerϕ]. ϕ∈HomR (M,C) For arbitrary modules B and M Im(B, M ) = P ϕ∈HomR (B,M ) Imϕ Le Phuong Thao Basic knowledge / 6 The property that B is a generator for Mod-R means that for any right R-module M, Im(B, M ) is as large as possible for every M and so equals M. For arbitrary modules C and M T Ker(M, C) = Kerϕ ϕ∈HomR (M,C) The property that CR is a cogenerator for Mod-R means that Ker(M, C) is as small as possible for every M and so equals 0. An R-module M is called a self-generator (self-cogenerator) if it generates all its submodules (cogenerates all its factor modules). Corollary 2.1.2 (a) If B is a generator and A is a module such that Im(A, B) = B, then A is also a generator; (b) Every module M such that there is an epimorphism from M to RR is also a generator; (c) If C is a cogenerator and D is a module such that Ker(C, D) = 0, then D is also a cogenerator. Generators and cogenerators can be characterized in the following theorem by properties of homomorphisms. Theorem 2.1.3 (a) B is a generator ⇔ ∀µ ∈ HomR (M, N ), µ 6= 0, ∃ϕ ∈ HomR (B, M ) : µϕ 6= 0. (b) C is a cogenerator ⇔ ∀λ ∈ HomR (L, M ), λ 6= 0, ∃ϕ ∈ HomR (M, C) : ϕλ 6= 0. 2.2 Injectivity and projectivity Injective modules may be regarded as modules that are ”complete” in the following algebraic sense: Any ”partial” 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 modules was considered. The ”injective” and ”projective” terminology was proposed in 1956 by Cartan and Eilenberg. Definition 2.2.1. Let M be a right R-module. (1) A submodule N of M is called essential or large in M if for any submodule X of M, X ∩ N = 0 ⇒ X = 0. If N is essential in M we denote N ⊂>∗ M. (2) A submodule N of M is called superfluous or small in M if for any submodule X of M, N + X = M, then X = M. In this case we write N ⊂>◦ M. (3) A right ideal I of a ring R is called a large right ideal of R if it is large in RR as a right R-module. Similarly, a right ideal I of a ring R is called a small right ideal of R if it is small in RR as a right R-module. (4) A homomorphism α : MR → NR is called large if Imα ⊂>∗ N. The homomorphism α is called small if Kerα ⊂>◦ M. Remark From the definition, we have the following: (1) A ⊂>◦ M ⇔ ∀U $ > M, A + U $ > M. (2) A ⊂>∗ M ⇔ ∀U ⊂> M, U 6= 0 ⇒ U ∩ A 6= 0. (3) M 6= 0 and A ⊂>◦ M ⇒ A 6= M. (4) M 6= 0 and A ⊂>∗ M ⇒ A 6= 0. Example 2.2.2 (1) For any module M, we have 0 ⊂>◦ M, M ⊂>∗ M. (2) A module M is called semisimple if every submodule is a direct summand. If M is a semisimple module, then only 0 is small in M and only M is essential in M. (3) In any free Z-module (free abelian group), only 0 is small. (4) Every finitely generated submodule of QZ is small in QZ . Le Phuong Thao Basic knowledge / 8 Lemma 2.2.3 ([63], Lemma 5.1.3) (1) A ⊂> B ⊂> M ⊂> N, B ⊂>◦ M ⇒ A ⊂>◦ N. n P Ai ⊂>◦ N. (2) Ai ⊂>◦ M, i = 1, 2, · · · , n ⇒ i=1 (3) A ⊂>◦ M and ϕ ∈ HomR (M, N ) ⇒ ϕ(A) ⊂>◦ N. (4) If α : A → B and β : B → C are small epimorphisms, then βα is also a small epimorphism. Lemma 2.2.4 ([63], Lemma 5.1.4) For a ∈ MR , the submodule aR of M is not small in M if and only if there exists a maximal submodule C ⊂> M such that a∈ / C. Lemma 2.2.5 ([63], Lemma 5.1.5) (1) A ⊂> B ⊂> M ⊂> N and A ⊂>∗ N ⇒ B ⊂>∗ M. n T (2) Ai ⊂>∗ M, i = 1, 2, · · · , n ⇒ Ai ⊂>∗ N. i=1 (3) B ⊂>∗ N and ϕ ∈ HomR (M, N ) ⇒ ϕ−1 (B) ⊂>∗ M. 4) If α : A → B and β : B → C are large monomorphisms, then βα is also a large monomorphism. Lemma 2.2.6 ([63], Lemma 5.1.6) Let A ⊂> MR . Then A ⊂>∗ MR ⇔ ∀m ∈ M, m 6= 0 ⇒ ∃r ∈ R : 0 6= mr ∈ A. Definition 2.2.7 Let M and U be two right R-modules. A right R-module U is said to be M-injective if for every monomorphism α : L → M and every 0 homomorphism ψ : L → U , there exists a homomorphism ψ : M → U such that 0 ψ α = ψ. Fac. of Grad. Studies, Mahidol Univ. 0 Ph.D. (Mathematics) / 9 α L - ψ p ? p pp pp M p pp 0 pp p ψ U A right R-module E is injective if it is M -injective, for all right Rmodule M. A right R-module M is called quasi-injective (or self-injective) if it is M -injective. The following Theorem gives us characterizations of injective modules. Theorem 2.2.8 ([63], Theorem 5.3.1) Let M be a right R-module. The following conditions are equivalent: (1) M is injective; (2) Every monomorphism ϕ : M → B splits (i.e. Im (ϕ) is a direct summand in B); (3) For every monomorphism α : A → B of right R-modules and any homomorphism ϕ : A → M, we can find a homomorphism ϕ : B → M such that ϕα = ϕ; (4) For every monomorphism α : A → B Hom(α, 1M ) : HomR (B, M ) → HomR (A, M ) is an epimorphism. A powerful test of injectivity is given as Baer’s Criterion which guarantees the equivalence between injectivity and R- injectivity. Theorem 2.2.9 ([100], 16.4) For a right R-module E, the following conditions are equivalent: (1) E is an injective R-module; (2) E is R-injective; Le Phuong Thao Basic knowledge / 10 (3) For every right ideal I of R and every homomorphism h : I → E, there exists y ∈ E with h(a) = ya, for all a ∈ I. Definition and basic properties of projective modules are dual to those of injective modules. Definition 2.2.10 A right R-module P is said to be M-projective if for every epimorphism β : M → N and every homomorphism ϕ : P → N , there exists a 0 homomorphism ϕ : P → M such that βϕ0 = ϕ. P ϕ ppp p pp 0 M p β p pp pp ϕ ? - N - 0 Now we have the following fundamental characterizations of projective modules. Theorem 2.2.11 ([63], Theorem 5.3.1) The following properties of a right Rmodule P are equivalent : (1) P is projective; (2) Every epimorphism ϕ : M → P splits (i.e. Ker(ϕ) is a direct summand in M); (3) For every epimorphism β : B → C of right R-modules and any homomorphism ϕ : P → C, there is a homomorphism ϕ : P → B such that βϕ = ϕ; (4) For every epimorphism α : B → C Hom(1P , β) : HomR (P, B) → HomR (P, C) is an epimorphism. Theorem 2.2.12 ([63], Theorem 5.4.1) A module is projective if and only if it is isomorphic to a direct summand of a free module. Fac. of Grad. Studies, Mahidol Univ. Ph.D. (Mathematics) / 11 Proposition 2.2.13 ([3], Proposition 16.10) Let M be a right R-module and (Uα )α∈A be an indexed set of right R-modules. Then (1) The direct sum L Uα is M -projective if and only if each Uα is M -projective. A (2) The direct product Q Uα is M -injective if and only if each Uα is M -injective. A Proposition 2.2.14 ([3], Corollary 16.11) Let (Uα )α∈A be an indexed set of right R-modules. Then (1) The direct sum L Uα is projective if and only if each Uα is projective. A (2) The direct product Q Uα is injective if and only if each Uα is injective. A 2.3 Noetherian and Artinian modules and rings Definition 2.3.1 (1) A right R-module MR is called Noetherian if every nonempty set of its submodules has a maximal element. Dually, a module MR is called Artinian if every set of its 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 with maximal condition and Artinian modules are called modules with minimal condition. Theorem 2.3.2 ([63], Theorem 6.1.2) Let M be a right R-module and let A be its submodule. I. The following statements are equivalent: (1) M is Artinian; Le Phuong Thao Basic knowledge / 12 (2) A and M/A are Artinian; (3) Every descending chain A1 ⊃ A2 ⊃ · · · ⊃ An−1 ⊃ An ⊃ · · · of submodules of M is stationary; (4) Every factor module of M is finitely cogenerated; (5) For every family {Ai | i ∈ I} 6= ∅ of submodules of M, there exists a finite subfamily {Ai | i ∈ I0 } (i.e., I0 ⊂ I and finite) such that T Ai = i∈I T Ai . i∈I0 II. The following conditions are equivalent: (1) M is Noetherian; (2) A and M/A are Noetherian; (3) Every ascending chain A1 ⊂ A2 ⊂ · · · ⊂ An−1 ⊂ An ⊂ · · · of submodules of M is stationary; (4) Every submodule of M is finitely generated; (5) For every family {Ai | i ∈ I} 6= ∅ of submodules of M, there exists a finite subfamily {Ai | i ∈ I0 } (i.e., I0 ⊂ I and finite) such that P Ai = i∈I P Ai . i∈I0 III. The following conditions are equivalent: (1) M is Artinian and Noetherian; (2) M is a module of finite length. The condition (I)(3) in Theorem 2.3.2 is called descending chain condition, briefly DCC. The condition (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. 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 2.4.1 A proper ideal P in a ring R is called a prime ideal of R if for any ideals I, J of R with IJ ⊂ P, then either I ⊂ P or J ⊂ P. An ideal I of a ring R is called strongly prime if for any a, b ∈ R with ab ∈ I, then either a ∈ I or b ∈ I. A ring R is called a prime ring if 0 is a prime ideal. (Note that a prime ring must be nonzero). Proposition 2.4.2 ([48], Proposition 3.1) For a proper ideal P of a ring R, the following conditions are equivalent: (1) P is a prime ideal; (2) If I and J are any ideals of R properly containing P , then IJ * P ; (3) R/P is a prime ring; (4) If I and J are any right ideals of R such that IJ ⊂ P, then either I ⊂ P or J ⊂ P ; (5) If I and J are any left ideals of R such that IJ ⊂ P, then either I ⊂ P or J ⊂ P ; (6) If x, y ∈ R with xRy ⊂ P, then either x ∈ P or y ∈ P. By induction, it follows from Proposition 2.4.2 that if P is a prime ideal in a ring R and J1 , . . . , Jn are right ideals of R such that J1 · · · Jn ⊂ P, then Ji ⊂ P for some i. By a maximal ideal in a ring we mean a maximal proper ideal, i.e., an ideal which is a maximal element in the collection of proper ideals. Proposition 2.4.3 ([48], Proposition 3.2) Every maximal ideal of a ring R is a prime ideal. Proposition 2.4.3 together with Zorn’s Lemma guarantees that every
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