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

From Sanh’s definition of prime submodules, we constructed some newnotions such as nilpotent submodules, nil submodules, a prime radical, a nil radicaland a Levitzki radical of a right o

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

PRIMENESS IN MODULE CATEGORY

Ms Le Phuong ThaoCandidate

Prof Banchong Mahaisavariya, Prof Yongwimon Lenbury,

M.D., Dip Thai Board of Orthopedics Ph.D

Faculty of Graduate Studies Doctor of Philosophy Program

Faculty of ScienceMahidol University

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 ThaoCandidate

Prof Banchong Mahaisavariya, Prof Skorn Mongkolsuk,

M.D., Dip Thai Board of Orthopedics Ph.D

Faculty of Graduate Studies Faculty of Science

I would like to express my sincere gratitude and appreciation to mymajor advisor, Dr Nguyen Van Sanh, for his constructive guidance, valuableadvice and inspiring talks throughout my study period that has enabled me tocarry 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 fromthe 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 ics, Mahidol University, for providing me with the necessary facilities and financialsupport Special thanks go to all the teachers and staffs of the Department ofMathematics for their kind help and support I would like to thank all of myfriends in the research group for their help throughout my study period at Mahi-dol University

Mathemat-I am very glad to express my thankful sentiment to Cantho Universityfor 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

PRIMENESS IN MODULE CATEGORY

LE PHUONG THAO 5137143 SCMA/D

Ph.D (MATHEMATICS)

THESIS ADVISORY COMMITTEE: NGUYEN VAN SANH, Ph.D MATICS), CHAIWAT MANEESAWARNG, Ph.D (MATHEMATICS), GUMPONSRITANRATANA, Ph.D (MATHEMATICS)

(MATHE-ABSTRACT

In modifying the structure of prime ideals and prime rings, many thors transfer these notions to modules There are many ways to generalize thesenotions and it is an effective way to study structures of modules However, fromthese notion definitions, we could not find any properties which are parallel tothat of prime ideals In 2008, N V Sanh proposed a new definition of a primesubmodule The definition was to let R be a ring, M a right R-module, and S beits endomorphism ring If any ideal I of S and any fully invariant submodule U of

au-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 foundmany beautiful properties of prime submodules that are similar to prime ideals

From Sanh’s definition of prime submodules, we constructed some newnotions such as nilpotent submodules, nil submodules, a prime radical, a nil radicaland a Levitzki radical of a right or left module M over an arbitrary associativering 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 thisresearch, we also transfered the Zariski topology of rings to modules

KEY WORDS : PRIME SUBMODULES/ ZARISKI TOPOLOGY

NILPOTENT SUBMODULES/ NIL SUBMODULESPRIME RADICAL/ NIL RADICAL/ LEVITZKI RADICAL

80 pages

Page

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 24

CHAPTER III 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

CHAPTER I INTRODUCTION

Throughout the text, all rings are associative with identity and all ules 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 endomor-phism ring, Mod-R for the category of all right R-modules and R-homomorphisms

mod-1.1 On the primeness of modules and submodules

Prime submodules and prime modules have been appeared in manycontexts Modifying the structure of prime ideals, many authors want to transferthis notion to right or left modules over an arbitrary associative ring By anadaptation 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 asmultiplication modules, but for the case of non-commutative rings, nearly we couldnot find something similar to the structure of prime ideals

In 1961, Andrunakievich and Dauns ([31], [71]) first introduced andinvestigated prime module Following that, a left R-module M is called prime iffor 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 ofprimeness, 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 primemodule if (0 :R M ) = A(M ), where A(M ) = {a ∈ R | aRm = 0, m ∈ M } For

the class of submodules, he also created the definitions of prime submodules andsemiprime submodules A submodule P of a left R-module M is called a primesubmodule if for any element r ∈ R and any element m ∈ M such that rRm ⊂ P,then either m ∈ P or r ∈ (P :RM ), and a submodule N of M is called a semiprimesubmodule if N 6= M and for any elements r ∈ R and m ∈ M such that rnm ∈ N,then rm ∈ N.

Following Bican ([20]), we say that a left R-module M is B-prime ifand only if M is cogenerated by each of its nonzero submodules It is easy to seethat B-prime implies prime In [100], it is pointed out that M is B-prime if andonly 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 Followinghim, a left R-module M is a strongly prime module if M is subgenerated by any ofits nonzero submodules, i.e., for any nonzero submodule N of M, the module Mbelongs to σ[N ], or equivalently, for any x, y ∈ M, there exists a set of elements{a1, · · · , an} ⊂ R such that annR{a1x, · · · , anx} ⊂ 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 thiscase, 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 aprime ideal of R and the submodule X is called a prime submodule if for r ∈ Rand m ∈ M with rm ∈ X implies either m ∈ X or r ∈ (X : M )

In 1993, McCasland and Smith ([4], [71], [74], [76]) gave a definitionthat a submodule P of a left R-module M is called a prime submodule if for anyideal 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 thestructure 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

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 andNoether in the 19th century, multiplication modules over commutative rings werenewly created by Barnard [9] in 1980 to obtain a module structure which behaveslike rings The structure of multiplication modules over noncommutative rings wasfirst 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 iffor any ideals I, J of R and any submodule X of M with IJ X ⊂ P, then either

IX ⊂ P or J X ⊂ 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 fullyinvariant submodule X of M is called a prime submodule if for any ideal I of S andany fully invariant submodule U of M, I(U ) ⊂ X implies I(M ) ⊂ X or U ⊂ X Afully invariant submodule is called semiprime if it equals an intersection of primesubmodules A right R-module M is called a semiprime module if 0 is a semiprimesubmodule of M Consequently, the ring R is semiprime ring if RR is a semiprimemodule By symmetry, the ring R is a semiprime ring if RR is a semiprime leftR-module

In 2008, Sanh ([87]) studied the concepts of M -annihilators and ofGoldie 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 dimensionand satisfies the ascending chain condition for M -annihilators A ring R is a rightGoldie ring if RR is Goldie as a right R-module It is equivalent to say that aring R is a right Goldie ring if it has finite right Goldie dimension and satisfiesthe ascending chain condition for right annihilators By using some properties ofprime modules and Goldie modules, we study the class of prime Goldie modules

1.2 On problems of primeness of modules and submodules

Recently, Sanh ([89], [90]) introduced the notions of nilpotent ules and nil submodules Let M be a right R-module and X, a submodule of M

submod-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 ule of M if IX is a right nil ideal of S From these new definitions, the authors alsointroduced prime radical, nil radical and Levitzki radical of a right R-module Mand investigated their properties in Chapter III and Chapter IV Another questionis: Can we construct and generalize of the Zariski topology of rings to modules byusing Sanh’s definition? The answer is positive in Chapter V of the thesis

submod-For the structure of the thesis, Chapter I is the introduction, Chapter

II contains basic knowledge, and main results are included in Chapters III, IVand V About the content of the study, Chapter I mentions preceding primenessconcepts in the module category which generalized the primeness in ring theory.Chapter II provides essential basic knowledge that is needed for the study ChapterIII deals with the formal definition, basic properties of nilpotent submodules of amodule 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 amodule The relation of prime radical, nil radical and Levitzki radical of a moduleare 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 inChapter VI

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 Mand 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 iffurther 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 ⊂>RR that I is a left ideal The notation I ⊂>R is reserved for two-sidedideals 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 animportant role in Module Theory and in some categories Below we will reviewthese notions

The property that B is a generator for Mod-R means that for any rightR-module M, Im(B, M ) is as large as possible for every M and so equals M.

For arbitrary modules C and M

ϕ∈Hom R (M,C)

Kerϕ

The property that CRis 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 ates all its submodules (cogenerates all its factor modules)

2.2 Injectivity and projectivity

Injective modules may be regarded as modules that are ”complete” inthe 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-phism (from all of B) into A.

Injective module first appeared in the context of abelian groups Thegeneral notion for modules was first investigated by Baer in 1940 The theory ofthese modules was studied long before the dual notion of projective modules wasconsidered The ”injective” and ”projective” terminology was proposed in 1956 byCartan 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 anysubmodule X of M, X ∩ N = 0 ⇒ X = 0 If N is essential in M we denote

(4) A homomorphism α : MR → NR is called large if Imα ⊂>∗N Thehomomorphism α is called small if Kerα ⊂>◦M

Remark From the definition, we have the following:

(1) For any module M, we have 0 ⊂>◦M, M ⊂>∗M

(2) A module M is called semisimple if every submodule is a directsummand If M is a semisimple module, then only 0 is small in M and only M isessential 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

(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 isalso a small epimorphism

Lemma 2.2.4 ([63], Lemma 5.1.4) For a ∈ MR, the submodule aR of M is notsmall in M if and only if there exists a maximal submodule C ⊂>M such that

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 everyhomomorphism ψ : L → U , there exists a homomorphism ψ0 : M → U such that

ψ0α = ψ

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 followingconditions are equivalent:

(4) For every monomorphism α : A → B

Hom(α, 1M) : HomR(B, M ) → HomR(A, M )

(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 everyepimorphism β : M → N and every homomorphism ϕ : P → N , there exists ahomomorphism ϕ0 : P → M such that βϕ0 = ϕ

homomor-(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 isisomorphic to a direct summand of a free module

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

A

Uα is M -projective if and only if each Uα is M -projective

(2) The direct productQ

A

Uα is M -injective if and only if each Uα is M -injective

Proposition 2.2.14 ([3], Corollary 16.11) Let (Uα)α∈A be an indexed set of rightR-modules Then

(1) The direct sum L

A

Uα is projective if and only if each Uα is projective

(2) The direct product Q

A

Uα is injective if and only if each Uα is injective

2.3 Noetherian and Artinian modules and rings

Definition 2.3.1 (1) A right R-module MR is called Noetherian if every nonemptyset of its submodules has a maximal element Dually, a module MR is calledArtinian if every set of its submodules has a minimal element

(2) A ring R is called right Noetherian (resp right Artinian) if themodule 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 andArtinian 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 itssubmodule

I The following statements are equivalent:

(1) M is Artinian;

(2) A and M/A are Artinian;

(3) Every descending chain A1 ⊃ A2 ⊃ · · · ⊃ An−1 ⊃ An ⊃ · · · ofsubmodules 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

(2) A and M/A are Noetherian;

(3) Every ascending chain A1 ⊂ A2 ⊂ · · · ⊂ An−1 ⊂ An ⊂ · · · ofsubmodules 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

(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 dition, briefly DCC The condition (II)(3) in Theorem 2.3.2 is called ascendingchain condition, briefly ACC Thus, Theorem 2.3.2 asserts that a module M isNoetherian if it satisfies ACC, and Artinian if it satisfies DCC

con-Corollary 2.3.3 ([63], con-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 everyfinitely generated right R-module MR is Noetherian (resp Artinian)

(3) Every factor ring of right Noetherian (resp Artinian) ring is againright Noetherian (resp Artinian).

2.4 Primeness in module category

In this section, before stating our new results we would like to list somebasic properties from [48]

Definition 2.4.1 A proper ideal P in a ring R is called a prime ideal of R if forany ideals I, J of R with IJ ⊂ P, then either I ⊂ P or J ⊂ P An ideal I of aring 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 ringmust be nonzero)

Proposition 2.4.2 ([48], Proposition 3.1) For a proper ideal P of a ring R, thefollowing conditions are equivalent:

(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, , Jnare right ideals of R such that J1· · · Jn⊂ P, then Ji ⊂ Pfor some i By a maximal ideal in a ring we mean a maximal proper ideal, i.e., anideal 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 aprime ideal

Proposition 2.4.3 together with Zorn’s Lemma guarantees that every

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 primering, 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 prime ideal

Theorem 2.4.6 ([48], Theorem 3.4) In a right or left Noetherian ring R, there existonly finitely many minimal prime ideals, and there is a finite product of minimalprime 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 anintersection of prime ideals (By convention, the intersection of the empty family

of prime ideals of R is R, so R is a semiprime ideal of itself) A ring R is called asemiprime ring if 0 is a semiprime ideal

Remark In Z, the intersection of any infinite number of prime ideals is 0 Theintersection of any finite list p1Z, , pkZ of prime ideals, where p1, , pk are dis-tinct prime integers, is the ideal p1· · · pkZ Hence the nonzero semiprime ideals of Zconsist of Z together with the ideals nZ, where n is any square-free positive integer

It follows from Proposition 3.6 [48] that an ideal I in a commutativering R is semiprime if and only if, whenever x ∈ R and x2 ∈ I, it follows that

x ∈ I The example of a matrix ring over a field shows that this criterion fails

in the noncommutative case However, there is an analogous criterion due toLevitzki-Nagata, as we will see in the next theorem

Theorem 2.4.8 ([48], Theorem 3.7) An ideal I in a ring R is semiprime if andonly if

(?) whenever x ∈ R with xRx ⊂ I, then x ∈ I

The reader should be aware that many authors define semiprime ideals

by the condition (?) in Theorem 2.4.8 From that view point, the theorem then

says that an ideal is semiprime if and only if it is an intersection of prime ideals.

Corollary 2.4.9 ([48], Corollary 3.8) For an ideal I in a ring R, the followingconditions are equivalent:

(1) I is a semiprime ideal;

(2) If J is any ideal of R such that J2 ⊂ I, then J ⊂ I;

(3) If J is any ideal of R such that J % I, then J2

* I;

(4) If J is any right ideal of R such that J2 ⊂ I, then J ⊂ I;

(5) If J is any left ideal of R such that J2 ⊂ I, then J ⊂ I

Corollary 2.4.10 ([48], Corollary 3.9) Let I be a semiprime ideal in a ring R If

J is a right or a left ideal of R such that Jn⊂ I for some positive integer n, then

J ⊂ I

Definition 2.4.11 An element x in a ring R is called a nilpotent element if xn = 0for some n ∈ N A right or a left ideal I in a ring R is called a nilpotent ideal if

In = 0 for some n ∈ N More generally, I is called a nil ideal if each of its elements

is nilpotent The prime radical P (R) of a ring R is the intersection of all the primeideals of R

Remarks ([48], page 53) (1) In Noetherian rings, all nil one-sided ideals are tent

nilpo-(2) If R is the zero ring, it has no prime ideals, and so P (R) = R If

R is nonzero, it has at least one maximal ideal, which is prime by Lemma 2.4.3.Thus, the prime radical of a nonzero ring is a proper ideal

(3) A ring R is semiprime if and only if P (R) = 0 In any case, P (R)

is the smallest semiprime ideal of R, and because P (R) is semiprime, it containsall nilpotent one-sided ideals of R

Now, let R be a semiprime ring and let A and B be right ideals of Rwith AB = 0, then (BA)2 = 0 and (A ∩ B)2 = 0, so that BA = 0 and A ∩ B = 0.Thus if I is an ideal of R then Ir(I) = 0 so that r(I)I = 0 Similarly, Il(I) = 0.Therefore l(I) = r(I) If I is a right annihilator then I = r(l(I)) = l(r(I)) so that

is also a left annihilator, and in these circumstances we call I an annihilator ideal

We have the following lemmas.

Lemma 2.4.12 ([100], Proposition 3.13) For a ring R with identity, the followingconditions are equivalent:

(1) R is a semiprime ring (i.e., P (R) = 0);

(2) 0 is the only nilpotent ideal in R;

(3) For ideals I, J in R with IJ = 0 implies I ∩ J = 0

Lemma 2.4.13 ([53], Lemma 1.16) Let R be a semiprime ring with the ACC alently DCC) for annihilators ideals, then R has only finite number of minimalprime ideals If P1, · · · , Pnare the minimal prime ideals of R then P1∩· · ·∩Pn= 0.Also a prime ideal of R is minimal if and only if it is an annihilator ideal

(equiv-Proposition 2.4.14 ([48], page 54) In any ring R, the prime radical equals theintersection of the minimal prime ideals of R

Definition 2.4.15 Let X be a subset of a right R-module M The right annihilator

of X is the set rR(X) = {r ∈ R : xr = 0 for all x ∈ X} which is a right ideal of

R If X is a submodule of M, then rR(X) is a two-sided ideal of R Annihilators

of subsets of left R-modules are defined analogously, and are left ideals of R If

M = R, then the right annihilator of X ⊂ R is

We now give the following basic properties of right and left annihilatorswhich have important consequences

Properties 2.4.16 ([53]) Let R be a ring and let X, Y be subsets of R Then wehave the following properties:

(1) X ⊂ Y implies that r(X) ⊃ r(Y ) and l(X) ⊃ l(Y );

endomor-By definition, the class of all fully invariant submodules of M is nonemptyand closed under intersections and sums Indeed, if X and Y are fully invariantsubmodules of M, then for every f ∈ S, we have f (X +Y ) = f (X)+f (Y ) ⊂ X +Yand f (X ∩ Y ) ⊂ f (X) ∩ f (Y ) ⊂ X ∩ Y In general, if {Xi : i ∈ I} where I is anindex set, is a family of fully invariant submodules of M, thenP

i∈I

Xi and T

i∈I

Xi arefully invariant submodules of M Especially, a right ideal I of a ring R is a fullyinvariant submodule of RR if it is a two-sided ideal

Now, let I, J ⊂ S and X ⊂ M For convenience, we denote I(X) =P

Definition 2.4.18 Let M be a right R-module and X, a fully invariant propersubmodule of M Then X is called a prime submodule of M (we say that X is prime

in M ) if for any ideal I of S, and any fully invariant submodule U of M, I(U ) ⊂ Ximplies I(M ) ⊂ X or U ⊂ X A fully invariant submodule X of M is called stronglyprime if for any f ∈ S and any m ∈ M, f (m) ∈ X implies f (M ) ⊂ X or m ∈ X

The following theorem gives some characterizations of prime ules similar to that of prime ideals and we use it as a tool for checking the primeness

submod-Theorem 2.4.19 ([86], [87]) Let M be a right R-module and P, a proper fullyinvariant submodule of M Then the following conditions are equivalent:

Examples 2.4.20 (1) Let Z4 = {0, 1, 2, 3} be the additive group of integers modulo

4 Then X =< 2 > is a prime submodule of Z4

(2) If M is a semisimple module having only one homogeneous nent, then 0 is a prime submodule Especially, if M is simple, then 0 is a primesubmodule

compo-Definition 2.4.21 A prime submodule P of a right R-module M is called a imal prime submodule if it is minimal in the class of prime submodules of M

min-The following proposition gives us a property similar to that of rings(see Lemma 2.4.5)

Proposition 2.4.22 [86] If P is a prime submodule of a right R-module M, then

P contains a minimal prime submodule of M

Lemma 2.4.23 [86] Let M be a right R-module and S = EndR(M ) Suppose that

X is a fully invariant submodule of M Then the set IX = {f ∈ S | f (M ) ⊂ X} is

a two-sided ideal of S

Theorem 2.4.24 [86] Let M be a right R-module, S = EndR(M ) and X, a fullyinvariant submodule of M If X is a prime submodule of M, then IX is a primeideal of S Conversely, if M is a self-generator and if IX is a prime ideal of S, then

X is a prime submodule of M

Definition 2.4.25 A fully invariant submodule X of a right R-module M is called

a semiprime submodule if it is an intersection of prime submodules of M

A right R-module M is called a prime module if 0 is a prime submodule

of M A ring R is a prime ring if RR is a prime module

A right R-module M is called a semiprime module if 0 is a semiprimesubmodule of M Consequently, the ring R is a semiprime ring if RRis a semiprimemodule By symmetry, the ring R is semiprime ifRR is a semiprime left R-module.Examples 2.4.26 (1) Every semisimple module with only one homogeneous com-ponent is a prime module Especially, every simple module is prime

(2) Every semisimple module is semiprime

(3) As a Z-module, the module Z4 is not semiprime

Theorem 2.4.27 [86] Let M be a prime module Then its endomorphism ring S

is a prime ring Conversely, if M is a self-generator and S is a prime ring, then

M is a prime module

Lemma 2.4.28 [86] Let M be a quasi-projective module, P be a prime submodule

of M, A ⊂ P be a fully invariant submodule of M Then P/A is a prime submodule

of M/A

Lemma 2.4.29 [86] Let M be a quasi-projective module and A a fully invariantsubmodule of M If ¯P ⊂ M/A is a prime submodule of M/A, then ν−1( ¯P ) is aprime submodule of M

For a right R-module M, let P (M ) be the intersection of all primesubmodules of M By our definition, M is a semiprime module if P (M ) = 0 Wewant to get some properties similar to that of prime radical of rings and at firststep, the following theorem is true for quasi-projective modules.

Theorem 2.4.30 [86] Let M be a quasi-projective module Then M/P (M ) is asemiprime module, that is, P (M/P (M )) = 0

Theorem 2.4.31 [86] If M is a semiprime module, then S is a semiprime ring

For the converse part of Theorem 2.4.31, we need M to be a projective, self-generator and finitely generated module that follows:

quasi-Proposition 2.4.32 [87] Let M be a quasi-projective, finitely generated right module which is a self-generator If S is a semiprime ring, then M is a semiprimemodule

R-Theorem 2.4.33 [86]

(1) If M is a prime module, then so is Mn for any n ∈ N

(2) If M is a semiprime module, then so is Mn for any n ∈ N

Proposition 2.4.34 ([88]) Let M be a quasi-projective, finitely generated rightR-module which is a self-generator Then we have the following:

(1) If X is a minimal prime submodule of M, then IX is a minimal prime ideal

R-(1) X is a semiprime submodule of M ;

(2) If J is any ideal of S such that J2(M ) ⊂ X, then J (M ) ⊂ X;

(3) If J is any ideal of S properly containing X, then J2(M ) 6⊂ X;

(4) If J is any right ideal of S such that J2(M ) ⊂ X, then J (M ) ⊂ X;

(5) If J is any left ideal of S such that J2(M ) ⊂ X, then J (M ) ⊂ X

From Theorem 2.4.35, we have the following corollary

Corollary 2.4.36 ([88]) Let M be a quasi-projective, finitely generated right module which is a self-generator and X, a semiprime submodule of M If J is aright or left ideal of S such that Jn(M ) ⊂ X for some positive integer n, then

(3) M is semiprime and satisfies the DCC on M -cyclic submodules

We next introduce the concept of Goldie dimension (also known asuniform dimension) of a module

Definition 2.4.38 A nonzero module M is said to be uniform if any two nonzerosubmodules of M have nonzero intersection, i.e., if every nonzero submodule of M

is essential in M

Let M be a right R-module Then M is said to have finite Goldiedimension if M does not contain a direct sum of a infinite number of nonzero sub-modules It is easy to show that M has finite Goldie dimension if M is Noetherian

or Artinian A ring R is said to have finite right Goldie dimension if R has finiteGoldie dimension as a right R-module

The next lemma gives the basic properties of modules of finite Goldiedimension

Lemma 2.4.39 ([53], Lemma 1.9) Let M be a nonzero right R-module.

(1) If M has finite Goldie dimension, then every nonzero submodule of

M contains a uniform submodule of M and there is a finite number of uniformsubmodules of M whose sum is direct and essential in M

(2) Suppose that M has uniform submodules U1, , Un such that thesum U1+ · · · + Unis direct and essential in M, then M has finite Goldie dimensionand the positive integer n is independent of the choice of Ui We call n the Goldiedimension of M and is denoted by dim(M )

Let M be a module of finite Goldie dimension Then by definition,submodules of M also have finite Goldie dimension, but it is not always true thatarbitrary factor modules of M have finite Goldie dimension For example, Q hasGoldie dimension 1 as a Z-module but Q/Z does not have finite Goldie dimension

If V is a vector space, then V has finite Goldie dimension if and only if V has finitedimension in the usual sense of linear algebra and in these circumstances, the twodimensions are equal

Proposition 2.4.40 [87] Let M be a quasi-projective, finitely generated right module which is a self-generator Then M has finite Goldie dimension if and only if

R-S has finite right Goldie dimension Moreover, in this case, dim(MR) = dim(SS).Definition 2.4.41 Let M be a right R-module A submodule X of M is called

an M -annihilator if X = Ker(I) = T

f ∈I

Ker(f ) for some subset I of S

We call M a Goldie module if M has finite Goldie dimension and satisfiesthe ACC on M -annihilators A ring R is called a right Goldie ring if RRis a Goldiemodule, or equivalently, if R has finite right Goldie dimension and satisfies the ACC

on right annihilators A right noetherian ring is right Goldie, but the converse isnot true

Lemma 2.4.42 [87] Let M be a right R-module and S = EndR(M ), its morphism ring If M satisfies the ACC (resp DCC) on M -annihilators, then Ssatisfies the ACC (resp DCC) on right annihilators

endo-Theorem 2.4.43 [87] Let M be a quasi-projective, finitely generated right module which is a self-generator If M is a Goldie module, then S is a rightGoldie ring.

R-Proposition 2.4.44 [87] Let M be a right R-module with finite Goldie dimensionand f ∈ S, a monomorphism Then f (M ) is an essential submodule of M

Definition 2.4.45 The right singular ideal of a ring R is denoted and defined by

Zr(R) = {x ∈ R | xK = 0 for some essential right ideal K of R}

In other words, if x ∈ R, then x ∈ Zr(R) if and only if rR(x) is anessential right ideal of R If Zr(R) = 0, then R is called a right nonsingular ring

Let M be a right R-module An element x ∈ M is said to be a singularelement of M if the right ideal rR(x) is essential in RR The set of all singularelements of M is called the singular submodule of M and is denoted by Z(M )

If Z(M ) = M, then M is called a singular module and if Z(M ) = 0, then M isnonsingular A ring R is right nonsingular if the right R-module RRis a nonsingularmodule

Theorem 2.4.46 ([68], Lemma 7.2) Let M be a right R-module

(1) Z(M ) · soc(RR) = 0, where soc(RR) denotes the socle of RR.(2) If f : M → N is any R-homomorphism, then f (Z(M )) ⊂ Z(N ).(3) If X ⊂ M, then Z(X) = X ∩ Z(M )

Proposition 2.4.47 ([87]) Let M be a quasi-projective, finitely generated rightR-module which is a self-generator If X is an essential submodule of M, then

IX = {f ∈ S | f (M ) ⊂ X} is an essential right ideal of S

Proposition 2.4.48 ([87]) Let M be a nonsingular right R-module with finiteGoldie dimension Then M satisfies the ACC and DCC on M -annihilators Espe-cially, if R is a right nonsingular ring with finite Goldie dimension, then R satisfiesthe ACC and DCC on right annihilators

Proposition 2.4.49 ([87]) Let M be a nonsingular right R-module with the ACC

on M -annihilators and let f ∈ S be such that f (M ) is an essential submodule of

M Then f is a monomorphism.

Definition 2.4.50 Let M be a right R-module The set of all prime submodules of

M is called the prime spectrum of M and denoted by Spec(M ) Recall that the set

of all prime ideals of R is called the prime spectrum of R and denoted by Spec(R)

or XR The topological structure on Spec(R) will help us to determine a topology

on Spec(M ) There are some useful facts about this topology on Spec(R)

Let R be a ring Denote Spec(R) (or XR) for the set of all prime ideals

of R For any ideal I of R, we define:

Let Γ(R) = {VR(I) | I is an ideal of R} From (1) − (3), there exists

a topology, say ΓR, on Spec(R) having Γ(R) as the family of all closed sets Thistopology is called the Zariski topology on Spec(R)

2.5 On Jacobson radical, prime radical, nil radical and Levitzki radical

of rings

Definition 2.5.1 Let M be a right R-module The Jacobson radical of M , denoted

by J (M ), is defined to be the intersection of all maximal submodules of M

In case M = R, we have J (RR) = J (RR) (by [63], Theorem 9.3.2) So

condi-(1) R is a semiprime ring;

(2) P (R) = 0;

(3) R has no nonzero nilpotent ideal;

(4) R has no nonzero nilpotent left ideal

Proposition 2.5.3([48]) Let R be a ring Then any semiprime ideal of R willcontain all nilpotent one-sided ideal of R

Since the prime radical of R is a semiprime ideal of R, we have thefollowing corollary

Corollary 2.5.4 ([48]) The prime radical of R contains all nilpotent one-sidedideals of R

Proposition 2.5.5 ([95], Proposition XV.1.4) If R satisfies ACC on two-sidedideals, 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, theJacobson radical is coincided with the prime radical

Theorem 2.5.7 ([48], Theorem 3.11) Let R be a right or left Noetherian ring andlet P be the prime radical of R Then P is a nilpotent ideal of R containing all thenilpotent right or left ideals of R

Theorem 2.5.8 (Hopkins-Levitzki theorem) ([48], Theorem 4.15) If R is a rightArtinian ring, then R is also right Noetherian and J (R) is nilpotent

In this case, we have J (R) = P (R)

Proposition 2.5.9 ([63], Corollary 9.3.7) For any ring R, we have the following:

(1) The sum of two nilpotent right, left or two-sided ideals is againnilpotent

(2) If RR is Noetherian, then every two-sided nil ideal is nilpotent.Proposition 2.5.10 ([84], Lemma 5.1) Let I be a nil ideal of R

(1) If J/I is a nil ideal of R/I, then J is a nil ideal of R;

(2) An arbitrary sum of nil ideals is nil

Definition 2.5.11 Let R be an arbitrary ring Then its nil radical N (R) is thesum of all nil two-sided ideals of R.

From proposition 2.5.10, we see that N (R/N (R)) = 0

Theorem 2.5.12 ([63], Theorem 9.3.8) Every (one-sided or two-sided) nil ideal iscontained in J (R)

Proposition 2.5.13 ([69], Proposition 10.27) For any ring R, we have P (R) ⊂

N (R) ⊂ J (R) If R is left Artinian, then P (R) = N (R) = J (R)

Theorem 2.5.14 (Levitzki’s theorem) ([69], Theorem 10.30) Let R be a rightNoetherian ring Then every nil one-sided ideal N of R is nilpotent We have

P (R) = N (R), and this is the largest nilpotent right (resp., left) ideal of R

Now, we will review the concept of locally nilpotent

Definition 2.5.15 Let I be a right ideal of a ring R I is called locally nilpotent

if for any finite subset {s1, · · · , sn} ⊂ I, there exists an integer k such that anyproduct of k elements from {s1, · · · , sn} is zero

Proposition 2.5.16 ([69] Proposition 10.31) Let I, J be locally nilpotent one-sidedideals in R Then I + J is locally nilpotent

Definition 2.5.17 The Levitzki radical of a ring R, denoted by L(R), is the sum

of all locally nilpotent ideals of R It is the largest locally nilpotent ideal of R, andcontains every locally nilpotent one-sided ideal of R Moreover, we have:

Proposition 2.5.18 ([69]) P (R) ⊂ L(R) ⊂ N (R) ⊂ J (R)

3.1 Prime submodules and semiprime submodules

It was shown in [46], Theorem 3.4, that there exist only finitely manyminimal prime ideals in a right Noetherian ring R Using this result we can provethe following theorem

Theorem 3.1.1 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator If M is a Noetherian module, then there exist onlyfinitely many minimal prime submodules

Proof Since M is a quasi-projective Noetherian module which is a self-generator,

it would imply that S is a right Noetherian ring Indeed, suppose that we have anascending chain of right ideals of S, I1 ⊂ I2 ⊂ · · · says Then we have I1(M ) ⊂

I2(M ) ⊂ · · · is an ascending chain of submodules of M Since M is a Noetherianmodule, there is an integer n such that In(M ) = Ik(M ) , for all k > n Then by([100], 18.4), we have In = Hom(M, In(M )) = Hom(M, Ik(M )) = Ik Thus thechain I1 ⊂ I2 ⊂ · · · is stationary, and hence S is a right Noetherian ring ByTheorem 2.4.6, S has only finitely many minimal prime ideals, P1, , Pt says By

Proposition 2.4.34, P1(M ), , Pt(M ) are the only minimal prime submodules of

Lemma 3.1.2 Let M be a quasi-projective, finitely generated right R-module which

is a self-generator and X, a simple submodule of M Then IX is a minimal rightideal of S

Proof Let I be a right ideal of S such that 0 6= I ⊂ IX Then I(M ) is a nonzerosubmodule of M and I(M ) ⊂ X Thus I(M ) = X and it follows from ([100], 18.4)

Proposition 3.1.3 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator Let X be a simple submodule of M Then either I2

X = 0

or X = f (M ) for some idempotent f ∈ IX

Proof Since X is a simple submodule of M, by Lemma 3.1.2, IX is a minimal rightideal of S Suppose that I2

X 6= 0 Then there is a g ∈ IX such that gIX 6= 0 Since

gIX is a right ideal of S and gIX ⊂ IX, we have gIX = IX by the minimality of IX.Hence there exists f ∈ IX such that gf = g The set I = {h ∈ IX | gh = 0} is aright ideal of S and I is properly contained in IX since f 6∈ I By the minimality of

IX, we must have I = 0 It follows that f2− f ∈ IX and g(f2− f ) = 0, and hence

f2 = f Note that f (M ) ⊂ X and f (M ) 6= 0, and from this we have f (M ) = X Corollary 3.1.4 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator Let X be a simple submodule of M If M is a semiprimemodule, then X = f (M ) for some idempotent f ∈ IX

Proof Since M is a semiprime module, it follows from Theorem 2.4.31 that S is

a semiprime ring and hence IX2 6= 0 Thus X = f (M ) for some idempotent f ∈ IX

Proposition 3.1.5 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator Then Zr(S)(M ) ⊂ Z(M ) where Zr(S) is the singularright ideal of S and Z(M ) is a singular submodule of M

Proof Let f ∈ Zr(S) and x ∈ M We will show that f (x) ∈ Z(M ) Since

f ∈ Zr(S), there exists an essential right ideal K of S such that f K = 0 It wouldimply that f K(M ) = 0 Note that K is an essential right ideal of S, we can seethat K(M ) is an essential submodule of M, and hence the set x−1K(M ) = {r ∈

R | xr ∈ K(M )} is an essential right ideal of R, and therefore f (x)(x−1K(M )) =

f (x(x−1K(M )) ⊂ f K(M ) = 0, proving that f (x) ∈ Z(M ) 

The following Corollary is a direct consequence of the above proposition

Corollary 3.1.6 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator If M is a nonsingular module, then S is a right nonsin-gular ring

Proposition 3.1.7 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator If M is a semiprime module with the ACC for M -annihilators, then M has only a finite number of minimal prime submodules If

P1, , Pn are minimal prime submodules of M, then P1 ∩ · · · ∩ Pn = 0 Also aprime submodule P of M is minimal if and only if IP is an annihilator ideal of S.Proof It follows from Theorem 2.4.31 that S is a semiprime ring Since Msatisfies the ACC for M -annihilators, we can see that S satisfies the ACC for rightannihilators (cf Lemma 2.4.42) By Lemma 2.4.13, S has only a finite number

of minimal prime ideals Therefore M has only finite number of minimal primesubmodules, by Proposition 2.4.34 If P1, , Pn are minimal prime submodules

of M, then IP1, , IPn are minimal prime ideals of S Thus IP1∩ · · · ∩ IPn = 0, byLemma 2.4.13 From IP1∩ · · · ∩ IPn = IP1∩···∩P n, we have P1∩ · · · ∩ Pn= 0 Finally,

a prime submodule P of M is minimal if and only if IP is a minimal prime ideal

of S It is equivalent to the fact that IP is an annihilator ideal of S Proposition 3.1.8 Let M be a quasi-projective right R-module and X be a fullyinvariant submodule of M Then the following are equivalent:

(1) X is a semiprime submodule of M

(2) M/X is a semiprime module

(2) ⇒ (1) By assumption, 0 is a semiprime submodule of M/X We

Q i ⊂ > M/X,Q i prime

ν−1(Qi) Since each Qi is a prime submodule of M/X, ν−1(Qi) is aprime submodule of M by Lemma 2.4.29 Therefore X is a semiprime submodule

3.2 Prime radical and nilpotent submodules

In this section, we introduce the notion nilpotent submodules and studythe properties of prime radical of a given right R-module M

Definition 3.2.1 Let M be a right R-module and X, a submodule of M We saythat X is a nilpotent submodule of M if IX is a nilpotent right ideal of S

From the definition we see that X is a nilpotent fully invariant module of M if and only if IX is a nilpotent two-sided ideal of S First, we get aproperty of semiprime submodules similar to that of semiprime ideals

sub-Proposition 3.2.2 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator and N be a semiprime submodule of M Then N containsall nilpotent submodules of M

Proof Let X be a nilpotent submodule of M Then IX is a nilpotent right ideal of

S Thus In

X = 0 for some positive integer n, and therefore In

X(M ) = 0 ⊂ N Since

N is a semiprime submodule of M which is a self-generator, X = IX(M ) ⊂ N,

Corollary 3.2.3 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator Let P (M ) be the prime radical of M Then P (M ) con-tains all nilpotent submodules of M

Proposition 3.2.4 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator If M satisfies the ACC on fully invariant submodules,then P (M ) is nilpotent.

Proof Since M satisfies the ACC on fully invariant submodules, S satisfies theACC on two-sided ideals Indeed, if I1 ⊂ I2 ⊂ · · · is an ascending chain of two-sided ideals of S, then I1(M ) ⊂ I2(M ) ⊂ · · · is an ascending chain of fully invariantsubmodules of M Since M has the ACC on fully invariant submodules, there exists

a positive integer n such that In(M ) = Ik(M ) for all k > n Thus In = Ik for all

k > n, showing that S satisfies the ACC on two-sided ideals Therefore P (S) isnilpotent (by Proposition 2.5.5) Since P (S) = IP (M ), we have P (M ) is nilpotent

no nonzero nilpotent submodules

Proof By hypothesis, 0 is a semiprime submodule of M If X is a nilpotentsubmodule of M, then In

X = 0 for some positive integer n, and hence In

X(M ) = 0.Note that IX(M ) = 0 by Corollary 2.4.36, and we can see that X = 0

Conversely, suppose that M contains no nonzero nilpotent ideals Let

J be an ideal of S such that J2(M ) = 0 Then we can write J = IJ (M ) andhence I2

J (M ) = 0 It follows that J (M ) is a nilpotent submodule of M and we get

J (M ) = 0 Thus 0 is a semiprime submodule of M by Theorem 2.4.35, showing

Proposition 3.2.6 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator Let P (M ) be the prime radical of M If M is Noetherianmodule, then P (M ) is nilpotent submodule of M containing all nilpotent submod-ules of M

Proof Let F = {P | P is a minimal prime submodule of M }

and G = {I | I is a minimal prime ideal of S} We write N = T

X∈F

X.Since N is semiprime, N contains all nilpotent submodules of M By proposition2.4.34, IN = T

assump-0, say P1· · · Pn= 0 Since IN is contained in each Pi, i = 1, , n; we have In

N = 0

Lemma 3.2.7 Let M be a right R-module and X, a fully invariant submodule of

M Then we have the following:

(1) For any n ≥ 1, lS(IX) ⊂>∗SS if and only if lS(In

X) ⊂>∗SS(2) If X is nilpotent, then lS(IX) ⊂>∗SS

Proof (1) We have lS(IX) ⊂ lS(IXn) So if lS(IX) ⊂>∗SS then lS(IXn) ⊂>∗SS versely, it suffices to show that lS(I2

Con-X) ⊂>∗SS implies lS(IX) ⊂>∗SS Let f be a nonzeroelement of S Since lS(IX2) ⊂>∗SS, there exists g ∈ S such that 0 6= f g ∈ lS(IX2) If

f gIX = 0 then f g ∈ lS(IX), so f S ∩ lS(IX) 6= 0 and we are done If f gIX 6= 0,then there is h ∈ IX such that f gh 6= 0 Then f ghIX ⊂ f gI2

X = 0 and hence

f gh ∈ lS(IX) which implies that f S ∩ lS(IX) 6= 0 Thus lS(IX) ⊂>∗SS

(2) If X is nilpotent, then IXn = 0 for some positive integer n It followsthat lS(In

X) = lS(0) = S Therefore lS(IX) ⊂>∗SS Proposition 3.2.8 Let M be a quasi-projective, finitely generated right R-modulewhich is a self-generator and assume that every ideal essential in SS contains aright regular element of S Then M is a semiprime module

Proof Let J be any ideal of S such that J2(M ) = 0 We wish to show that

J (M ) = 0 Since J2(M ) = 0 and J = IJ (M ) we see that X := J (M ) is nilpotent.Thus lS(IX) = lS(J ) is an ideal which is essential in SS (by Lemma 3.2.7) Hence

lS(J ) contains a right regular element f of S Now from f J = 0 and f is rightregular, we have J = 0 Thus J (M ) = 0, proving that 0 is semiprime submodule

of M Hence M is a semiprime module Proposition 3.2.9 Let M be a right R-module and X, a nilpotent submodule of

M If N is a fully invariant submodule of M with M = N + IX(M ), then M = N

In particular, if M = IX(M ), then M = 0

Proof We prove by induction on i ≥ 1 that M = N + IXi (M ) The case n = 1

is true by the hypothesis Suppose M = N + Ii

X(M ) Then IX(M ) = IX(N ) +

IXi+1(M ) Thus M = N + IX(M ) = N + IX(N ) + IXi+1(M ) = N + IXi+1(M ) Since

X is nilpotent, IX is nilpotent Then In

X = 0 for some positive integer n, and so

IXn(M ) = 0 Thus M = N In particular, 0 is a fully invariant submodule of M

Proof (1) Let K be a right ideal of S and IX is properly contained in K Then X

is properly contained in K(M ) Thus K(M ) = M by the maximality of X, provingthat K = S

(2) Since I is a right ideal of S, we see that X := I(M ) is a submodule

of M and I = IX If N is a submodule of M and X is properly contained in N,then IX is properly contained in IN Therefore IN = S by the maximality of I

Let M be a right R-module The Jacobson radical of M , denote J (M ),

is the intersection of all maximal submodules of M The Jacobson radical of thering R is the intersection of all maximal right ideals of R

Recall that if R is right Artinian, then J (R) is the largest nilpotentright, left or two-sided ideal of R ([63], Corollary 9.3.10) Hopkins-Levitzki Theo-rem says that if the ring R is right Artinian, then R is right Noetherian and the