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A STUDY ON UNISERIAL AND SERIAL RINGS AND MODULES
by Md. Aftab Uddin Student No.1009093006 P, Registration No. 1009093006 Session: October 2009
A dissertation submitted in partial fulfillment of the requirements for the award of the degree of MASTER OF PHILOSOPHY in Mathematics
Department of Mathematics BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY (BUET) DHAKA-1000 March 2016 ii
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CANDIDATE’S DECLARATION
It is hereby declared that this thesis or any part of it has not been submitted elsewhere for the award of any degree or diploma.
MD. AFTAB UDDIN
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DEDICATED TO MY PARENTS AND TEACHERS
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ACKNOWLEDGEMENT
I am highly grateful to my supervisor Dr. Khandker Farid Uddin Ahmed, Professor, Department of Mathematics, BUET, Dhaka-1000, for his invaluable suggestions, constant inspiration and supervision that has enabled me to complete my thesis. I am also grateful to the honorable member of the Board of Examiners Professor Dr. Md. Mustafa Kamal Chowdhury and Professor Dr. Md. Abdul Alim for their thoughtful and constructive comments in preparing and writing this thesis. I would like to express my thankful sentiment to the Head of the Department Professor Dr. Md. Abdul Alim for providing me with all kinds of generous help and support during the period of my study in this Department. I am highly grateful to all the respected teachers and staffs of the Department of Mathematics, BUET, Dhaka-1000. I wish to thanks my external examiner of the Board of Examiners Dr. Sujoy Chakraborty, Professor, Department of Mathematics, Shahjalal University of Science and Technology (SUST), Sylhet-3114. A special recognition goes to the former head of the department Prof. Dr. Md. Manirul Alam Sarker who encouraged me to continue the program. Finally, I would like to express my special gratitude to my employing and controlling authority Ministry of Education (MoE) and Directorate of Secondary and Higher Education (DHSE), Dhaka-1000, respectively for giving me the permission to perform the course. I express my thankfulness to all my Principals, especially to my present Principal Professor Md. Abul Kalam Azad and Shiba Prashad Das Gupta, Head of the Department, Department of Mathematics, Feni Government College, Feni, for extending their support and co-operation during the period of my study. Special thanks to my beloved wife, our two children for their amour, devoutness, helping attitude and support to me in this regard. Above all, I am grateful to Almighty Allah who has given me the strength and patience to finish the program.
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MD. AFTAB UDDIN ABSTRACT
A right R-module M is called uniserial if its submodules are linearly ordered by inclusion, i.e., for any submodules A and B of M, either A⊆ B or B ⊆ A. A ring R is right uniserial if it is uniserial as a right R-module. A right R-module M is called a serial module if it is a direct sum of uniserial modules. A ring R is right serial if it is serial as a right R-module and R is serial if R is left and right serial. Modifying some structures of uniserial and serial rings over associative arbitrary rings, present study develops some properties of uniserial and serial modules over associative endomorphism rings. Some characterizations of Noetherian (resp. Artinian) uniserial and serial modules over endomorphism rings are also investigated in the present study.
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CONTENTS Page No. Title page i Certification page ii Declaration page iii Dedication page iv ACKNOWLEDGEMENT v ABSTRACT vi CHAPTER I INTRODUCTION 1 CHAPTER II BASIC KNOWLEDGE 4 2.1 Preliminaries 4 2.2 Modules and Submodules 11 2.3 Local Rings and Local Modules 22 2.4 Prime and Semiprime Rings and Modules 24 2.5 Noeherian and Artinian Rings and Modules 28 2.6 Radical and Socle of Modules 33 2.7 Injective and Projective Modules 36 CHAPTER III UNISERIAL AND SERIAL RINGS 41 CHAPTER IV UNISERIAL AND SERIAL MODULES 49 CONCLUSION 66 REFERENCES 67
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CHAPTER I INTRODUCTION
Ring theory plays a vital role in the field of pure and applied Mathematics. The elliptic curves are used in algebraic geometry and number theory, both of which study commutative rings. In abstract algebra, ring theory is the study of rings, the algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, modules, special classes of rings for example, group rings, division rings, universal enveloping algebras, as well as an array of properties that proved to be of interest both within the theory itself and for its applications such as homological properties and polynomial identities. Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by representing their elements as linear transformations of vector spaces and studies modules over these abstract algebraic structures.
Background and present state of the problem Ring theory is an indispensable part of Algebra. It is widely applied in electrical and Computer Engineering [33]. Module theory appears as a generalization of theory of vector space over field. Every field is a ring and every ring may be considered as a module. A module is called uniserial if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings proved that every finitely generated module over a serial ring is a direct sum of uniserial modules. He also investigated the structure of modules over a serial ring and show that the endomorphism ring of a projective module over a serial ring is a local serial ring. Finally he proved that any two simple modules over an indecomposable serial ring have the same endomorphism rings [10]. In 1991, Huynh D. V. and Dan P. [15] proved that a ring satisfying the conditions “Every finitely generated right R-module is a direct sum of projective module with zero socle and uniserial Artinian modules’’ is serial Noetherian or serial Artinian. They also proved that a
2 right non-singular rings satisfying the above condition is a direct sum of uniserial Artinian ring, a serial ring with zero right socle and prime rings. In 1998, Dung N. V. and Facchini A., [8] showed that every direct sum of serial module is serial. They also proved that every direct summand of a finite direct sum of copies of a uniserial module is again a direct sum of copies of the module. They investigated the properties of uniserial modules using arbitrary direct sum and finite direct sum. In 2003, Sanh.N.V. and Chotchaisthit [22] characterized right p-injectivity when the ring R is right uniserial or serial by considering p-injectivity on the classes of uniserial and serial modules. In 2004, Singh S. and Al-Bleehed H. [26] proved that an Artinian ring R is generalized uniserial if and only if every indecomposable right R-module is uniserial. Every uniserial module is local. They also investigated the matrix representation of such rings. It is well-known that every serial Noetherian ring satisfies the Restricted Minimum Conditions (RMC), such a ring is a direct sum of an Artinian rings and hereditary prime rings. In 2006, Somsup C. and Sanh N.V. [27] proved that every serial ring having RMC is Noetherian. In 2006, Prihoda P. [21] gave some results on uniserial modules which are not quasi- small. He proved N0-th root property for uniserial modules and showed that the Krull- Schmidt theorem hold for direct sum of uniserial modules that are not quas-small. He also showed that if a right R-module M is finite direct sum of uniserial modules then any direct sum of M is uniserial. In 2008, Soonthornkrachang, Dan P., Sanh N. V. and Shum K. P. [28] proved that a right perfect ring R is serial Artinian if and only if every right ideal is a direct sum of a projective module and a singular uniserial module. They also gave some characterization of right Harada rings and serial Artinian rings. In 2010, Singh S., [25] proved if R is a right Artinian ring such that every finitely generated, indecomposable right R -module is local then R is a direct sum of local modules. He also proved that if M is right module over a right Artinian ring such that any finitely generated submodules of any homomorphic image of M is a direct sum of local modules, then it is a direct sum of local modules.
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In 2012, Sompong S. [29] gave some properties that can imply the right R-module M to be serial. In general, every epimorphic image of a serial module need not to be serial, but he gave some conditions for an epimorphic image of serial module to be serial. Next some properties of serial Artinian right R-modules have been investigated. In this thesis, Chapter I deals with early brief history of uniserial and serial rings and modules. All basic definitions, examples, counter examples and properties of related topics are included in Chapter II. Definitions and examples of uniserial and serial rings and their properties are given in Chapter III. Also some modified structures of uniserial and serial rings are included here. Finally, the structure of uniserial and serial modules and their modified structures are given in Chapter IV.
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CHAPTER II BASIC KNOWLEDGE
Overview Throughout this thesis, all rings are associative with identity and all modules are unitary right R-modules. We denote by R an arbitrary ring and by mod-R, the category of all right
R-modules. The notation M R indicates the right R-module M, when 1R is assumed to be unital (i.e. to have the property that .1 mm for any Mm . The set NMHom ),( denotes the set of right R-module homomorphism between two right R-module M and N and if further emphasis is needed, the notation R NMHom ),( is used. The kernel of any
R NMHomf ),(( is denoted by fKer )( and the image of f by f ).Im( In particular
R MEnd )( denotes the ring of endomorphism of a right R-module M.
A submodule X of M is indicated by writing XM. Also, I MR means that I is a right ideal of R. The notation IR is reserved for ideal i.e., two-sided ideal. As usual , ,,and represent the set of natural numbers, integers, rational numbers, real numbers and complex numbers respectively.
2.1 Preliminaries Before dealing with deeper results on the structure of rings with the help of module theory, we provide first some essential elementary definitions, examples and properties.
DEFINITIONS Ring A ring ( R +,*) is a set R together with two binary operations ‘+’ and ‘*’ which we call addition and multiplication defined on R such that the following axioms are satisfied : 1. ( R ,+) is an abelian group.
2. Multiplication is associative i.e. for any Rzyx ,,, we have zyxzyx )*()( 3. For any Rzyx ,,, the left distributive law zxyxzyx )*()()( and the right distributive law xzyxxzy )()()( hold. Our ring will be an associative ring with identity. A ring ( R ,+,*) is called commutative if the multiplication operation is commutative.
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Also, if there exists a multiplicative identity in the ring, that is, an element e such that for all a in R, a e e a a , then it is said to be a ring with unity. The number 1 is a common example of a unity.The ring in which e is equal to the additive identity must have only one element. This ring is called the trivial ring. Rings that sit inside other rings are called subrings. Maps between rings which represent the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category called the category of rings.
Examples (i) , , , are the ring.
(ii) Let be a ring and let M n () be the collection of all n n matrices having elements of as entries. The operations of addition and multiplication in R allow us to add and multiply matrices in usual fashion and so M n () is a ring. Also M n (), M n (), M n (), are ring. (iii) Let F be the all function PPf .: We know that (F,+) is an abelian group under the usual function addition by xgxfxgf )()())(( and function multiplication by
xgfxfg ))(())(( , that is fg is a function whose values at x is xgxf )()( and so F is ring.
Division ring or skew field In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. More formally, a ring with 0 ≠ 1 is a division ring if every non-zero element a has a multiplicative inverse. Division rings differ from fields only if their multiplication is not required to be commutative. The condition 10 is only there to exclude the trivial ring with a single element .10 Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements. All fields are division rings; more interesting examples are the non-commutative division rings. If we allow only rational instead of real coefficients in the constructions of the quaternion, we obtain another division ring. In general, if R is a ring and M is a simple module over R, then the endomorphism ring of M is a division ring; every division ring arises in this fashion from some simple module.
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Examples (i) , , and are three division rings, since non-zero elements of each of these rings form a multiplicative group.
Factor ring or quotient ring Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R/I={ : RaIa } together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
Boolean ring A ring in which every element is idempotent is a boolean ring.
Commutative ring A ring R is commutative if the multiplication is commutative, i.e., rs sr for all , Rsr .
Example , , and are all commutative rings.
Ring with zero divisors A ring R is called with zero divisors if there exists at least two elements a and b of R such that ab=0, where a 0 and b 0 .
Example In the ring of residue classes modulo 6 i.e.,
6 ={ 5,4,3,2,1,0 }, we have ,03..2 but 02 and 03 . Thus 6 is a ring with zero divisors.
Ring without zero divisors A ring R is called without zero divisors if the product of two non-zero elements of R is not zero, i.e., for all a, b in R, ab=0 a 0 or b 0 or both a 0 and b 0 .
Examples (i) , , and are rings without zero divisors. (ii) All rings of residue classes modulo of any prime number are rings without zero divisors. Thus 5 is a ring with zero divisors.
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Integral domain or entire ring A commutative ring D with unity and without zero divisors is called an integral domain.
Examples (i) , , and are integral domains.
(ii) In the ring of integers the residue classes modulo 6 , i.e., 6 is a ring but not an integral domain. (iii) The ring of even integers is not an integral domain, since it does not contain the unit element.
Field A ring F with at least two elements is said to be a field if it satisfies the following conditions: (i) F is a commutative ring; (ii) F has unity, i.e., there exists ,.11.01 FaaaaF .
(iii) F has multiplicative inverse, i.e., aFa 0, , there exists 1 Fa such that
11 Faaaaaa .,
Example The set of integers is an integral domain but not field, since a a 1 .
Local ring A ring R is called local if it has exactly one maximal element. A ring with only finitely many maximal ideals is called semi local ring.
Examples (i) Any field is a local ring, since {0} is its only maximal ideal. (ii) In an ideal ,...}4,3,2,1,0{ all are maximal ideal except {0}. Hence is semi local.
Prime ring A ring R is said to be a prime ring if any two ideal I and J of R, such that IJ={0}, implies I={0} or J={0}.
Examples (i) Any domain is a prime ring.
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(ii) Any simple ring is a prime ring. (iii) Every primitive ring is a prime ring.
Primitive ring A left primitive ring is a ring that has a faithful simple left R-module. Every simple ring is primitive. Primitive rings are prime.
Semisimple ring The decompositions of simple ring is called a semisimple ring.
Example (i) Every Noetherian ring is semisimple. (ii) Every nilpotent ideals having no nilpotent elements is semisimple. 2 000 (mod 3) 12 1 1(mod 3) 22 4 1(mod 3)
In 3 (integer modulo 3), 1 and 2 are simple ring.
Simple ring A ring R is called a simple ring if it has no proper ideals. A non-zero ring with non-zero two-sided ideals is a simple ring.
Examples (i) ,and are three division rings, since non-zero elements of each of these rings form a multiplicative group. (ii) Every division ring is a simple ring.
Trivial ring The ring consisting only of a single element {0} is called trivial ring.
Ring with unity A ring R is called ring with unity if there exists an element 10 R such that Raaaa .,.11.
Examples (i) , , , are all rings with unity.
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Unique factorization domain or factorial ring A unique factorization domain is defined to be an integral domain R in which every non- zero element x of R can be written as a product of prime elements of R. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements. Unique factorization domains are often called factorial ring.
Examples (i) All principal ideal domains are called unique factorization domain. (ii) All Euclidean domains are called unique factorization domain.
Zero ring A ring in which the product of any two elements is 0 (the additive neutral element). R={0} is a zero ring as 000 R , 00.0 R , i.e. R is a zero ring.
Ideal Let R be a ring. Then a subring I of R is called an ideal of R if it is left ideal as well as right ideal. That is, a subring I is called an ideal of R if r R and Ii implies Iri , Iir .
Examples (i) I= {… -4, -2, 0, 2, 4, …} is an ideal of ={… -3, -2, -1, 0, 1, 2, 3, …}
ba 0 b (ii) The set R dcba Z,,,: is a ring and I ca Z,: be a dc 0 d subring of R, then I is an ideal of R.
Prime Ideal Let R be a ring and let I be a ideal of R. Then I is called prime ideal of R if Iab implies either Ia or Ib .
Example (i) ={… -3, -2, -1, 0, 1, 2, 3, …} I=5 ={… -15, -10, -5, 0, 5, 10, 15, …} J= 10={… -30, -20, -10, 0, 10, 20, 30, …}
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Here, I is prime ideal, since )25(10 I , where 5 I but 2 I , But )65(30 J , where neither 5 nor 6 not belongs to 30.
Definition (Krull) A proper ideal P in a ring R is called a prime ideal of R if for any I, J of R such that IJ P , then I P or J P.
Principal ideal An ideal I of a ring R is called Principal ideal of R if the ideal is generated by a single element of R.
Examples (i) Suppose, = {… -3, -2, -1, 0, 1, 2, 3, …} is an ideal. Then I={… -3m, -2m, -1m, 0, 1m, 2m, 3m, …} is a principal ideal of R since it generated by a single element m. (ii) = {… -3, -2, -1, 0, 1, 2, 3, …} is itself a principal ideal, since it is generated by a single element 1.
Maximal ideal Let R be a commutative ring. Then an ideal I of R is called maximal if there is no proper ideal of R which is properly contained in I.
Example (i) = {… -3, -2, -1, 0, 1, 2, 3, …}
I1= {… -15, -10, -5, 0, 5, 10, 15, …}
I2= {… -30, -20, -10, 0, 10, 20, 30, …}
Here, I1 is the maximal ideal of , since the only ideal of which properly contained in I1 is itself.
But, I2 is not the maximal ideal of , since I2 is not properly contained in as I1= {… -15, -10, -5, 0, 5, 10, 15, …} is properly contained in .
Composition series A composition series for a module M is a chain of submodules
0 MM 10 ... n MM
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such that each of the factors / MM ii 1 is a simple module. The number of gaps (namely n) is called the length of the series and the factors / MM ii 1 are called the composition factors of M corresponding to this composition series. By convention, the zero module is considered to have a composition series of length zero with no composition factors. A module of finite length is any module which has a composition series and a module M has finite length if and only if M is both Noetherian and Artinian.
Let V = VK be vector space and let {x1, x2, …, xn} be a basis of V.
n1 n Then, 0 x1K x1K +x2K … i Kx i Kx =V i1 i1 is a composition series of V. does not have a composition series because does not contain a simple ideal and the module does not have a composition series because
contains neither a minimal (=simple) nor a maximal submodule. The -module /6has two composition series. 0 2/6/6, 0 3/6/6, The factors of the first are 2/6 /3, (/6)/(/6) /2 and those of the second are 3/6 /2, (/6)/(/6) /3 2.2 Modules and submodules
Definition Let R be a ring with identity and M an abelian group. This M is called a right R-module if there exists ),(, mrmrMMR satisfying the following conditions:
(i) , 1 Mmm and 1 )( 1rmmrrmmRr
(ii) Mm and , 1 1 )( mrmrrrmRrr 1
(iii) Mm and , 1 1 )()( rmrrrmRrr 1 (iv) Mm and 1.1 mmR Similarly, we define left R-modules by operate to the left side of M. we can write
M R RM )( to indicate that M is a right R-module. Let M be an R-module. A subset L of M is a submodule of M if L is an additive subgroup and Lm , LmrRr , i.e., L is a
12 module under operations inherited from M. when L is a submodule of M, we can define the quotient module (factor module) / LM with the operation / / LMRLM given by (i) ),( mrrm (ii) LmrrLmrLm .)(),( Let R, S be two rings and let M be an abelian group. Then M is called an R-S bimodule if M is a left R-module, right S-module, and if for any rR, mM, sS, we have r (ms) =
(rm) s. We denote it by M SR . Let M be a module and mM. The element m generates a cyclic submodule m X of M. There is an epimorphism : mXX given by (a) = ma and ker = {am|a = 0} = Ann(m), the annihilator of m. Hence m X X / Ann(m). A subset I of a ring R is a right ideal if (i) ,, IyxIyx (ii) ,, IxrRrIx The ring R can be considered as a right R- module, every right ideal can be considered as a submodule of RR . If X is fully invariant submodule then
X XR XMfSfIM })(:{: is a two sided ideal.
If M is right-R module: X XR XMfSfIM })(:{: is a right ideal
If M is left-R module: X R X XMfSfIM })(:{: is a left ideal.
Examples (i) Every ring R is an R-module over itself. (ii) Let R be a ring and let M be a left ideal of R, then M is an R-module. (iii) Every abelian group is a module over the ring of integers.
Definition A right R-module M is called a semiprime module if 0 is a semiprime submodule of M.
Consequently, a ring R is called semiprime ring if RR is a semiprime module. A fully invariant submodule is called a semiprime submodule if it is an intersection of prime submodules.
Definition Let M and N be R-modules. A map : NMf is a homomorphism if
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(i) , 1 Mmm and 1 mfmfmmfRr 1 ).()()( (ii) Mm and rmfmrfRr .)()(
Proposition 2.2.1 [2] Let M and N be rings and : NMf be a ring homomorphism. Then (a) f is onto N if and only if Nf ;)Im( (b) f is an injective if and only if fKer ;0)( If : NMf is R-linear, we define its kernel as mfMmfKer }0)(:{)( and its image as Mmmff }:)({)Im( , Ker(f) is a submodule of M, and Im(f ) is a submodule of N, f is called a monomorphism if mfmf 1 )()( 1 Mmm .,
Definition
An R-homomorphism : R NMf R is called (1) a monomorphism if for any X Mod-R and for any homomorphism ,:, ghfogfohMXgh (2) an epimorphism if for any X Mod-R and for any homomorphism ,:, ghgofhofXNgh (3) an isomorphism if f is a monomorphism and epimorphism.
Proposition 2.2.2[2] Let M and N be left R-modules and let : NMf be an R-homomorphism. Then the following statements are equivalent: (a) f is an epimorphism onto N; (b) Nf ;)Im(
(c) For every R K and every pair :, KNhg of R-homomorphism, hfgf implies hg ;
(d) For every R K and every R-homomorphism KNg ,: gf 0 implies g .0
Proposition 2.2.3 [2] Let M and N be left R-modules and let : NMf be an R-homomorphism. Then the following statements are equivalent:
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(a) f is a monomorphism ; (b) fKer ;0)(
(c) For every R K and every pair :, MKhg of R-homomorphism, ghfg implies hg ;
(d) For every R K and every R- homomorphism MKg ,: g : K M implies hg ;
Definition Let M be a right R-module. A homomorphism : MMf is called an endomorphism.
The abelian group R MMHom ),( becomes a ring if we use the composition of maps as multiplication. This ring is called the endomorphism ring of M and we denote it by
R MEnd ).(
Proposition 2.2.4 [2] Let R and S be a rings and M an abelian group. If M is a left R-module via : l MEndRf )( and a right S-module via : MEndSg )( then the following are equivalent :
(a) M SR ;
(b) MEndRf S )(: is a ring homomorphism;
(c) R MEndSg )(: is a ring homomorphism.
Definition
Let M be a right R-module and R MEndS )( its endomorphism ring. A submodule X of M is called a fully invariant submodule of M if for any Ss , we have XXs .)( By the definition, the class of all fully invariant submodules of M is non-empty and closed under intersections and sums. Especially, a right ideal of R is a fully invariant submodule of RR if it is a two-sided ideal of R. Let , SJI and MX . For convenience, we denote
XfXI )()( , fKerIKer )()( and iiii ,1,,|{ nniJyIxyxIJ }. If If 1 ni With these notations, we can see that for any right R-module M and any right ideal I of R, the set MI is a fully invariant submodule of M.
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Definition
Let M be a right R-module and MEndS R )( . Suppose that X is a fully invariant submodule of M. Then the set X XMfSfI })(:{ is a two sided ideals of S. By the definition, the class of all fully invariant submodules of M is nonempty and closed under intersections and sums. Indeed, if X and Y are fully invariant submodules of M, then for every Sf , we have )()()( YXYfXfYXf and YXYfXfYXf .)()()(
In general, if i IiX }:{ where I is an index set, is a family of fully invariant submodules of M, then X i and X i are fully invariant submodules of M. Ii Ii
Proposition 2.2.5 [2] If R is a ring and and denote respectively the left and right multiplication, then
M MEndR R )(: and REndR )(: are ring isomorphism.
Definition Let M be a right R-module and X, a proper fully invariant submodule 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, if XUI ,)( then either )( XMI or XU . A fully invariant submodule X of M is called strongly prime if for any Sf and any Mm , if )( XMf , then either )( XMf or Xm .
Theorem 2.2.6 [23] Let M be a right R-module and P, a proper fully invariant submodule of M. Then the following conditions are equivalent: (a) P is a prime submodule of M; (b) For any right ideal I of S and any submodule U of M, if PUI ,)( then either )( PMI or PU ; (c) For any S and any fully invariant submodule U of M, if PU ,)( then either )( PM or PU ;
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(d) For any left ideal I of S and any subset A of M, if )( PMI or PAIS ,)( then either )( PMI or PA ; (e) For any S and any Mm , if MmPmS ,,))(( then either )( PM or Pm . Moreover, if M is quasi-projective, then the above conditions are equivalent to: (f) / PM is a prime module. In addition, if M is quasi-projective and a self-generator, then the above conditions are equivalent to: (g) If I is a ideal of S and U, a fully invariant submodule of M such that I(M) and U properly contain p, then PUI .)(
Definition The rin g R is semisimple (or completely reducible) if R is semisimple as a right R-module. A right ideal of R which is simple as an R-module is called a minimal right ideal. A semisimple ring is thus a direct sum of minimal right ideals, and every simple module is isomorphic to a minimal right ideal of R. The module 0 is semisimple as an empty sum of simple submodules but 0 is not a simple module, since it was assumed that for a simple module R, R 0. Every abelian group may be considered as a -module; so an abelian group is semisimple if it is a semisimple -module. The factor group/n, n 0, is a semisimple -module if and only if n is square-free (i.e., n is the product of pair wise distinct prime numbers, 1 ,..., ppn k or n 1). The modules and are not semisimple since they have no simple submodules.
Theorem 2.2.7 [17]
R Let R be s ring. Then RR is a semi-simple right R-module if and only if R is a semi- simple left R-module.
The following theorem gives some characterizations of semisimple submodules.
Theorem 2.2.8 [17] For a right R-module M, the following conditions are equivalent: (a) Every submodule of M is a sum of simple submodules; (b) M is a sum of simple submodules; (c) M is a direct sum of simple submodules;
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(d) Every submodule of M is a direct summand of M.
Proposition 2.2.9 [30] The following properties of a ring R are equivalent: (a) R is a semi-simple ring. (b) All right R-modules are semi-simple. (c) All right R-modules are projective. (d) All right R-modules are injective. (e) Every right ideal of R is a direct summand of R.
Definition If A is a submodule of a module M, then R is a direct summand of M if there exists a submodule B of M such that M A B and BA }.0{0 In this case we can write
BAM . The direct summands of RR correspond to idempotent elements of R, i.e., Re such that 2 ee .
Definition Let M be a right R-module and X, a subset of M. Then the set X ) is called the submodule of M-generated by X, where iii i ;,...,2,1,,:{)| nniRrXxrxX } and this is nii the smallest submodule of M containing X. A subset X of M R is called a free set (or linearly independent set) if for any 21 ,...,, k Xxxx and for any 21 k Rrrr ,,...,, we have
k ii i kirrx .},...,2,1{00 A subset X of M R is called a basis of M if M = X ) i1 and X is a free set. If a module M has a basis then M is called a free module.
Theorem 2.2.10 [30] (a) A right R-module M is free if and only if M is isomorphic to R I )( for some set I; (b) If R is a division ring, then every right R-module is free; (c) Every module is a quotient module of a free module.
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Definition A right R-module M is said to be finitely generated if there exists a finite set of generators for M, or equivalently, if there exists an epimorphism n MR for some n . In particular, M is cyclic if it is generated by a single element, or equivalently, if there exists an epimorphism MR . It follows that M is cyclic if and only if / IRM for some right ideal I of R. For example, let M be a right R-module and Mm , then m generates a cyclic submodule mR of M. There is an epimorphism : mMRf given by )( mrrf and mrRrfKer },0|{)( which is a right ideal of R. Hence fKerRmR ).(/
Lemma 2.2.11 [30] Let X be a submodule of a right R-module M. (a) If M is finitely generated, then so is M / X. (b) If X and M / X are finitely generated, then so is M.
Theorem 2.2.12 [17]
A right R-module M is finitely generated if and only if for any family i IiA }:{ of submodules Ai M with i MA , there exists a finite subfamily i IiA 0 }:{ Ii where 0 II and I 0 is finite, such that i MA . Ii
Examples
In 6 = { 5,4,3,2,1,0 } = 1), 6 is a -module. Then
1. 1) =6, 2) = ( 4,2,0 ) = 4), 3 ) = { 3,0 }, 3,2 ) =6, because x3 + y 2 = 1 for some
x, y .
2. { 2 } is not free because 023 , { 3,2 } is not free because 3 2 + 32 = .0
Hence 6 is a finitely generated 6 -module.
Theorem 2.2.13 [17] The following ststements hold: (1) Every free right R-module is isomorphic to R X )( for some set X. (2) Every right R-module is an epimorphic image of a free module.
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Note (i) A module M is called a free module if it has basis. (ii) Every free right R-module is isomorphic to R X . (iii) Every right R-module is an epimorphic image of a free module.
Definition A module M is simple (or irreducible) if M 0 and the only submodules of M are 0 and M. Every simple module M is cyclic, in fact it is generated by any non-zero Mx . It is clear that M is simple if and only if M /1 where is a maximal right ideal of I.
Proposition 2.2.14 [23] The following properties of an exact sequence 0 ZYX 0 are equivalent: (a) The sequence splits.
(b) There exists a homomorphism : XY such that X .1
(c) There exists a homomorphism : YZ such that Z.1
Theorem 2.2.15 [2] For a left R-module the following statements are equivalent: (a) M is semisimple; (b) M is generated by simple modules; (c) M is the sum of some set of simple submodule; (d) M is the sum of its simple submodules; (e) Every submodule of M is a direct summand; (f) Every short exact sequence 0 NML 0 of left R-modules splits.
Note (i) Every simple module is cyclic. (ii) M is simple ,0 mRMMm Definition
A module M R is called a semisimple module if and only if every submodule of M is a
20 direct summand, i.e., M is semi-simple if and only if for any submodule MX , there exists a submodule Y M such that YXM .
Proposition 2.2.16 [2] The following properties of a module M are equivalent: (a) M is semisimple. (b) M is a direct sum of simple modules. (c) Every submodule of M is a direct summand.
Lemma 2.2.17 [2]
If M is a simple module, then R MEnd )( is a skew-field.
Definition A submodule X of a right R-module M is called a simple submodule (or minimal submodule) if X is a simple module, i.e., X M is minimal M is nonzero and submodule X M , 0 X M MX .
Definition X is called a maximal submodule of M if X M and for any submodule Y M if X Y M , then Y X or Y M .
Theorem 2.2.18 [2] The following ststements hold: (a) Every finitely generated right R-module contains at least one maximal submodule. Therefore, every ring with identity contains at least one maximal right ideal. (b) For any submodule X M , X is maximal if and only if / XM is a simple module. (c) M is simple if and only if for any mRMMm .,0
Definition A submodule A of a right R-module M is called essential or large in M if for any nonzero submodule U of M, UA .0 If A is essential in M, we denote A * M . A right ideal I of a ring R is called essential if it is essential in RR . For any right R-module M, we always have R*M. Any finite intersection of essential submodules of M is again essential in M, but it is not true in general. For example, consider the ring of integers. Every nonzero
21 ideal of is essential in but the intersection of all ideals of is 0 which is not essential in . Since any two nonzero submodules of have nonzero intersection, is an essential extension of . A monomorphism : MU is said to be essential if Im( )*M. A*M 0 U M, UA .0 U 0, UAUM 0 U UAM 0, U .0
Proposition 2.2.19 Every nonzero ideal is essential in .
Proposition 2.2.20 [23] Let M be a right R-module. Then for any submodule AM, A*M mrRrmMm 0:,0, and Amr .
Proposition 2.2.21 For any MMod-R, let AB M. If A*M, then (i) A*B and (ii) B *M.
Proposition 2.2.22 * * Let A and B be essential submodules in M R .Then BA M and A B M.
Definition
A submodule A of M R is called superfluous or coessential or small in M if for any submodule U of M, MUA , implies U = M, or equivalently, MU , implies X+U M. A right ideal I of a ring R is called superfluous in R if it is a superfluous submodule of RR . Every module has at least one superfluous submodule, namely 0. The sum of a finite number of superfluous submodules of M is again a superfluous in M, but we are not sure about the arbitrary sum. For example, take as a -module. In , every cyclic submodule is superfluous but the sum q = which is not superfluous in . Qq
An epimorphism : NM is said to be superfluous if Ker( ) M. If A is superfluous
o o in M. We denote it by * MA i.e., * UMA , MUMUAM .
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2.3 Local rings and local modules
Definition An element r of a ring R is called right invertible (resp. left invertible) if there is an r R with rr 1( resp. rr 1). Then r is called the right inverse (resp. left inverse) of R. If rr rr 1, then r is said to be invertible and r is said to be inverse of r.
Theorem 2.3.1 [17] Let A be the set of all non-invertible elements of R, then the following conditions are equivalent:
(a) A is additively closed, i.e., , 21 21 AaaAaa ; (b) A is two sided ideal; (c) A is the largest proper right ideal; (d) A is the largest proper left ideal; (e) In R there exists a largest proper right ideal; (f) In R there exists a largest proper left ideal; (g) For every r R either r or 1-r is right invertible; (h) For every r R either r or 1-r is left invertible; (i) For every r R either r or 1-r is invertible;
Definition A ring is called local ring if it satisfies the following properties: (a) R has a unique maximal left ideal; (b) R has a unique maximal right ideal; (c) 01 and the sum of any two non-units in R is a non-unit; (d) 01 and for any r R , r and 1 r is a unit. Before 1960 many authors required that a local ring be (left and right Noetherian) and (possibly non- Noetherian ) local ring were called quasi- local rings.
Examples (i) All fields (and skew fields) are local rings since {0} is the only maximal ideal in these rings. (ii) A non-zero ring in which every element is either a unit or nilpotent is a local ring.
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(iii) The ring of rational numbers with odd denominator is local.
Corollary 2.3.2 [17] Let R be a local ring and I be the ideal of non-invertible elements of R. Then the following conditions are equivalent: (a) R/I is a skew field; (b) Every left invertible (resp. right invertible) element is invertible; (c) Every non-zero ring, which is the image of a local ring under a surjective ring homomorphism, is itself local; In particular, every isomorphic image of a local ring is local.
Proposition 2.3.3 If R is a ring and I is a prime ideal then S is a local ring with maximal ideal 1IS
Lemma 2.3.4 [Nakayama’s Lemma, version 1] Suppose that R is a local ring and M is a finitely generated R-module. If mM 0 then M=0.
Lemma 2.3.5 [Nakayama’s Lemma, version 2] Suppose that R is a local ring and E is a finitely generated R-module and EF is a submodule. If mEFE ,0 then E = F.
Proposition 2.3.6 [35] For a ring R the following properties are equivalent: (a) R is local; (b) There is a maximal left ideal which is superfluous in R; (c) The sum of two non-invertible elements in R is non-invertible; (d ) There is a maximal right ideal which is superfluous in R;
Definition A module M is called local if it has a largest proper submodule. By definition, we see that every local module is cyclic. The largest proper submodule is the unique maximal submodule but the converse is not true. For example, let M= 2. Since 2 is simple, we see that is maximal in M and it is the unique maximal submodule of M. But is
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not largest because 2 . From this fact, we conclude that is not finitely generated. Hence we see that if M has unique maximal submodule and M is finitely generated then M is local and consequently it is cyclic.
Proposition 2.3.7 [35] For a self-injective R-module M the following conditions are equivalent: a)( M is indecomposable; b)( Every non-zero submodule is essential in M;
c)( R MEnd )( is a local ring;
2.4 Prime and Semiprime rings and modules
Definition Let R be a ring and P be any proper ideal of R. Then there exists two ideal I and J of R with PIJ , either PI or PJ . A ring R is a prime ring if RR is a prime module, i.e.,
0 is a prime submodule of RR . A prime ring is a ring in which 0 is a prime ideal. Note that a prime ring must be non-zero.
Examples (i) Any domain is a prime ring. (ii) Any simple ring is a prime ring and more generally, every left or right primitive ring is a prime ring. (iii) Any matrix ring over an integral domain is a prime ring. In particular, the ring of 2 2 integer matrix is a prime ring.
Proposition 2.4.1 Let R be a prime ring. Then the following conditions are equivalent: (a) For any two ideals I and J of R, IJ }0{ implies I }0{ or J }.0{ (b) For any two right ideals I and J of R, IJ }0{ implies I }0{ or J }.0{ (c) For any two left ideals I and J of R, IJ }0{ implies I }0{ or J }.0{
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Proposition 2.4.2 [13] For a proper ideal P in a ring R, the following conditions are equivalent: (a) P is a prime ideal; (b) If I and J are any ideals of R properly containing P, then PIJ ; (c) / PR is a prime ring; (d) If I and J are any right ideals of R such that PIJ , then either PI or PJ ; (e) If I and J are any left ideals of R such that PIJ , then either PI or PJ ; (f) If , Ryx with PxRy , then either Px or Py .
Corollary 2.4.3 [23] For a proper ideal P in a ring R , the following conditions are equivalent: (a) P is a prime ideal; (b) If I and J are right ideals of R such that PIJ , either I P or PJ ; (c) For any Ra and any ideal I of R such that PaI , either PaR or PI ; (d) If I and J are left ideals of R such that PIJ , either I P or PJ ; (e) If , Ryx with PxRy , either Px or Py ; (f) PR is a prime ring;
Proposition 2.4.4 [35] For a ring R with unit, the following conditions are equivalent: (a) R is semiprime (i.e., RP 0)( ); (b) 0 is the only nilpotent (left) ideal in R; (c) For two ideals I and J in R with IJ 0 , also JI ;0\
Corollary 2.4.5 [13] For an ideal I in a ring R, the following conditions are equivalent: (a) I is a semi-prime ideal; (b) If J is any ideal of R such that 2 IJ , then IJ ;
(c) If J is any right ideal of R such that 2 IJ , then IJ ;
(d) If J is any left ideal of R such that 2 IJ , then IJ ;
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Definition A ring R is prime if 0 is a prime ideal. Thus, an ideal P is prime iff the ring / PR is a prime ring. More generally, an ideal I of R is semiprime if it is an intersection of prime ideals. A ring R is semiprime if 0 is a semiprime ideal of R. So an ideal I is semiprime iff R/I is a semiprime ring. The smallest of all the semiprime ideals, :{)( PPRN is a prime ideal of R} is the prime radical of R. So R is semiprime iff RN .0)(
Theorem 2.4.6 [2] For a ring R, the following conditions are equivalent: (a) R is semiprime; (b) RN ;0)(
(c) I 2 ,0 implies I 0 for every left (right/two-sided) ideal I of R; (d) aRa ,0 implies a ;0 (e) R has no non-zero nilpotent left (right/two-sided) ideals; (f) IJ ,0 implies JI 0\ for every pair I, J of left (right/two-sided) ideals;
Lemma 2.4.7 [18] For a ring R the following conditions are equivalent: (a) R is semisimple; (b) R is semiprime and right Artinian; (c) R is semiprime and satisfies the descending chain condition on principal right ideals.
Theorem 2.4.8 [35] Let M be a quasi-projective module, P be a prime submodule of M , A P be a fully invariant submodule of M . Then / AP is a prime submodule of AP ./
Definition 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.
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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 semiprime. By symmetry, the ring R is a semiprime ring if R R is a semiprime left R-module.
Examples (i) Every semisimple module with only one homogeneous component is a prime module. Especially, every simple module is prime. (ii) Every semisimple module is semiprime.
(iii) As a -module, a module 4 is not semiprime.
Theorem 2.4.9 [16] 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 2.4.10 [23]
Let M be a right R-module and i IiP },{ be a family of fully invariant submodules of M and put 0 PP i . Let i PMfSfI i })(|{ , i=0 or Ii .Then i II 0 Ii Ii
Theorem 2.4.11 [23] Let M be a right R-module and MEndS ).( If M is a semiprime module, then S is a semiprime ring.
Theorem 2.4.12 [23] (a) If M is a prime module, then so is M n for any Nn ;
(b) If M is a semiprime module, then so is M n for any Nn ;
Theorem 2.4.13 [23] Let M be a quasi projective module. Then MPM )(/ is a semiprime module, that is MPMP .0)(/)(
Corollary 2.4.14 [35] If R is a prime (resp. semiprime) ring, then R n is a prime (resp. semiprime) R-module and hence n RMat )( is a prime (resp. semiprime) ring.
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2.5 Noetherian and Artinian rings and Modules
Definition A ring R is called left Artinian if it satisfies the descending chain condition (DCC) on left ideals if every descending chain of left ideals LLL 321 ... becomes stationary after a finite number of steps, i.e. for some k , we get k 1 k 2 LLL k 3 ... Similarly , A ring R is called right Artinian if it satisfies the descending chain condition
(DCC) on right ideals if every descending chain of right ideals LLL 321 ... becomes stationary after a finite number of steps, i.e. for some k , we get k 1 k 2 LLL k 3 ... An Artinian ring is a ring which is left Artinian as well as right Artinian. The importance of this finiteness condition was first realized by Emil Artin.
Examples (i) A finite direct product of Artinian ring is Artinian. (ii) A quotient of an Artinian ring (by a two sided ideal) is Artinian. (iii) A finitely generated module over an Artinian ring is Artinian.
Definition A ring R is called left Noetherian if it satisfies the ascending chain condition (ACC) on left ideals if every ascending chain of left ideals LLL 321 ... becomes stationary after a finite number of steps, i.e. for some k , we get k 1 k 2 LLL k 3 ... Similarly , A ring R is called right Noetherian if it satisfies the ascending chain condition
(ACC) on right ideals if every descending chain of right ideals LLL 321 ... becomes stationary after a finite number of steps, i.e. for some k , we get k 1 k 2 LLL k 3 ... A Noetherian ring is a ring which is left Noetherian as well as right Noetherian . These rings first were investigated by Emmy Noether.
Examples (i) If R contains a subring (with unity) which is a division ring, and if R is finite dimensional as a left module over the division ring, then R is Artinian and Noetherian. (ii) Any principal ideal ring is Noetherian.
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Note (i) If a ring R is right Artinian, then R is right Noetherian but the converse is not true. For example, consider, m n / mn and
n m1 m2 m3 mmmm 2312 ,.../,/... . The chain 2 4 8 2... ... is not stationary. So is not Artinian. Thus is Noetherian but not Artinian. (ii) A right R-module M is Artinian but it need not be Noetherian. For example, the
a Purifier group P , where P ={ pa 1),(: and i }0{ } p and p i
p/=:P .
In P , there are infinite ascending chains of submodules, so P does not satisfy the
ACC and consequently, it is not Noetherian. The module is Noetherian but not Artinian. Indeed, since every ideal of is principal and therefore finitely generated, it is 2 Noetherian. Since the chain 2 2 ... is not stationary, we can conclude that
is not Artinian. The Purifier groupP = p/ is Artinian but not Noetherian.
Corollary 2.5.3 [35] Let : BA be a surjective map from the ring A to a ring B, then If the ring A is Noetherian then the ring B is Noetherian. ( If the ring A is Artinian then the ring B is Artinian.
Definition
A non-empty family of submodules of a right R-module M satisfies the Ascending
Chain Condition (ACC), i.e. if for every chain 21 ... n1 MMMM n ... of elements in , there is a positive integer n such MM nin for i= 1, 2, … A right R-module M is called Noetherian if and only if every nonempty family of submodules of M has a maximal element by inclusion. A ring R is called right (resp., left)
Noetherian if and only if RR (resp., R R ) is a Noetherian right (resp., left) R-module. The ring R is called Noetherian if and only if it is both right and left Noetherian.
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A non-empty family of submodules of a right R-module M satisfies the Descending
Chain Condition (DCC), i.e. if for every chain 1 2 .... n1 MMMM n ... of elements in , must terminate after a finite number of steps. A right R-module M is called Artinian if and only if every nonempty family of submodules of M has a minimal element by inclusion. A ring R is called right Artinian if RR is an Artinian module. The ring R is called Artinian ring if it is both right and left Artinian.
In p , there are infinite ascending chains of submodules , so p does not satisfy the
ACC and consequently it is not Noetherian. The module is Noetherian but not Artinian. Indeed, since every ideal of is principal ideal and therefore finitely generated. Since the 2 chain 2 2 … is not stationary. We conclude that is not Artinian. The prufer group p = p/ is Artinian but not Noetherian.
Proposition 2.5.4 [30] A module is Noetherian if and only if every strictly ascending chain of submodules is finite.
Proposition 2.5.5 [30] Let L be a submodule of a module M. Then M is Noetherian if and only if both L and / LM are Noetherian.
Proposition 2.5.6 [30] If A ring is Noetherian, then every finitely generated module is Noetherian.
Definition A right R-module M is called Artinian if every nonempty family of submodule has a minimal element by inclusion. A right R is called right- Artinian if RR is Artinian as a right R-module.
Theorem 2.5.7 [2] Let M be a right R-module and A M . Then the following conditions are equivalent: (a) M is Noetherian; (b) A and M / A are Noetherian;
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(c) Any ascending chain A1 A2 An … of submodules of M is stationary, i.e., there exists n such that AA nn .1 This condition is called the ascending chain condition or ACC. (d) Every submodule of M is finitely generated.
Theorem 2.5.8 [2] Let M be a right R-module and let A be its submodule. Then the following statements are equivalent: (a) M is Artinian; (b) A and M / A are Artinian;
(c) Any descending chain AA 21 An of submodules of M is stationary. This condition is called the descending chain condition or DCC. (d) Every factor module of M is finitely co-generated. ][0 AYYXfA 0 Let 0 ZYX 0 be an exact sequence of right R-modules. Then Y is Noetherian (resp. Aartinian) X and Z are Noetherian (resp. Artinian).
Corollary 2.5.9 [13] (1) The image of Artinian ( resp. Noetherian) module is also Artinian( resp. Noetherian ). (2) The finite sum of artinian ( resp. noetherian ) submodules of M is also Artinian (resp. Noetherian). (3) The finite direct sum of Artinian (resp. Noetherian) modules of M is also Artinian (resp. Noetherian). (4) If R is semi-simple, then R is both left and right Artinian (resp. Noetherian).
Lemma 2.5.10 [30] Let L be a submodule of a module M. Then the following conditions are equivalent: (a) M is finitely generated then so is M/L; (b) If L and M/L are finitely generated then so is M;
Theorem 2.5.11 [35] For any R-module M the following properties are equivalent:
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(a) M is Noetherian; (b) The set of finitely generated submodules of M is Noetherian; (c) Every non-empty set of finitely generated submodules of M has a maximal element; (d) Every submodule of M is finitely generated;
Definition A right R-module M is called locally Noetherian if every finitely generated submodule is Noetherian. The following gives some characterizations of locally Noetherian modules:
Theorem 2.5.13 [35] (a) Let 0 NML 0 be an exact sequence in a right R-module M. Then the following conditions are equivalent: (i) If M is (locally) Noetherian, then L and N are (locally) Noetherian; (ii) If L and N are Noetherian, then M is Noetherian; (iii) If L is Noetherian and N is locally Noetherian, then M is locally Noetherian; (b) The direct sum of locally Noetherin modules is locally Noetherian. Analogously, right R-module M is called locally Artinian if every finitely generated submodule is Artinian.
Theorem 2.5.14 [35] (a) Let 0 NML 0 be an exact sequence in a right R-module M. Then the following conditions are equivalent: (i) If M is (locally) Artinian, then L and N are (locally) Artinian; (ii) If L and N are Artinian, then M is Artinian; (iii) If L is Artinian and N is locally Artinian, then M is locally Artinian; (b) The direct sum of locally Artinian modules is locally Artinian. (c) If M is a locally Artinian right R-module, then (i) Every finitely generated right R-module in M ][ is Artinian; (ii) MRadM )(/ is a semisimple module.
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2.6 Radical and Socle of modules
Definition Let M be a right R-module. Then the sum of all superfluous submodules of M is called the radical of M and is denoted by MRad ).( It is also the intersection of all maximal submodule of M. If there is no maximal submodule of M then MMRad .)( For example,
Rad()=. For a ring R, we have Rad(RR)=Rad(RR) and by this fact we can define the J the Jacobson radical of a ring R by R R RRadRRadRJ )()()( as a two sided ideal of R. The radical of a module plays a vital role in the study of the structure of modules and rings. If M is finitely generated then Rad(M) is superfluous in M. The first important property is that Rad(M)/Rad(M) = 0. The following theorem gives one of the properties of radical of a module.
Proposition 2.6.1 [30] Let R be a ring.Then the following statements hold : (a) 1|{)( abRaRJ has a right inverse for all Rb }; (b) RJ )( is the largest two sided ideal of R consisting of elements a such that a)1( is invertible; (c) RJ )( is the intersection of all maximal ideals of R; (d) RJ )( is the intersection of all annihilators of simple R-modules; In general the ring RJR )(/ is not semisimple. For example, take R , we have J()=Rad()=0 and hence /J()=/0 , which is not simple.
Theorem.2.6.2 [2] Let M be a right R-module. Then M is finitely generated if and only if MRadM )(/ is finitely generated and the natural epimorphism MRadMM 0)(/ is superfluous, i.e. MRad )( o M.
Theorem.2.6.3 [2] Let M be a right R-module. Then the following conditions are equivalent: (a) MRad 0)( and M is Artinian; (b) MRad 0)( and M is finitely cogenerated;
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(c) M is semisimple and finitely generated; (d) M is semisimple and Noetherian; (e) M is a direct sum of finite set of simple submodules.
Theorem 2.6.4 [2] For right R-module M and N we have the following: (a) If : NM is an R-homomorphism, then NRadMRad )())(( . In particular, for any R MEndS ),( MRadMRad )())(( , i.e. MRad )( is fully invariant submodule of M. (b) MRadMRad 0))(/( for any submodule C M , if CMRad 0)/( then MRad )( C, i.e., MRad )( is the smallest submodule C of M such that CMRad 0)/( .
Theorem 2.6.5 [13] If M is finitely generated by right R-module and if MRMJ ,)( then M=0. This theorem is often called Nakayama’s Lemma. It implies in particular that if M is finitely generated right R-module and if 21 ,...,, mmm n are element of M such that the cossets i RMJm )( generates RMJM )(/ , then 21 ,...,, mmm n generate M.
Definition Let M be a right R-module. Then the sum (resp., direct sum) of all simple (minimal) submodule of M is called a Socle of M and is denoted by Soc(M). We also define the Socle of a module M as the intersection of all essential submodules of M. If MSoc 0)( , then M does not contain any simple submodules. For example, we have Soc( )=0, Especially )( MMSoc if and only if M is semisimple. For example, any vector space V over a field K is a semisimple K-module, since all the one dimensional subspace of V are simple K- submodules. In any ring R, RSoc R )( and R RSoc )( need not to be equal although they are two sided ideals. They coincide only if R is a semiprime ring.
Theorem 2.6.6 [2] Let M be a right R-module, then (a) M is finitely cogenerated if and only if MSoc )( is finitely cogenerated and the inclusion map )(0 MMSoc is essential, i.e. MSoc )( o M;
35
(b) M is finitely generated if and only if MSoc )( is essential and finitely generated.
Theorem 2.6.7 [17] Let M and N be right R-module, then the following conditions hold: (a) If : NM is an R-homomorphism, then NSocMSoc )())(( . In particular, for any R MEndS )( , MSocMSoc )())(( , i.e. MSoc )( is a fully invariant submodule of M. (b) MSocMSocSoc )())(( and for any submodule C M , CCSoc ,)( then C MSoc ),( i.e. MSoc )( is the largest submodule C of M which coincides with its Socle.
Definition The Socle of a module M is the sum of all simple submodules of M and is denoted by Soc(M). By convention the sum of the empty of submodules is the zero submodule. Hence Soc(M)=0 if and only if M has no simple submodules. A semisimple (or completely reducible ) module is any module M such that Soc(M)=0. Let R M be a module. Then its Socle is the sub module denoted by Soc(M) and is defined by :{)( NMNMSoc is simple}. So the Socle of M is the largest submodule of M generated by simple modules, or equivalently, it is the largest semisimple submodule of M. The radical of M is the submodule denoted by Rad(M) and is defined by :{)( NMNMRad is maximal in M}. So the radical of M is the smallest submodule of M modulo which M is cogenerated by simples, or equivalently, it is the smallest submodule modulo which M can be embedded in a product of simples.
Corollary 2.6.8 [17] For a left R-module M, the following conditions are equivalent : (a) M is semisimple Soc(M)= M and M is cosemisimple Rad(M)= 0; In particular, Soc(Soc(M)) = Soc(M) and Rad(M/Rad(M)) = 0;
Theorem 2.6.9 [13] For any ring R, the following conditions are equivalent: (a) All right R-modules are semi-simple; (b) All left R-module are semi-simple;
36
(c) RR is semi-simple;
(d) R R is semi-simple; (e) Either R is the zero ring or DMDMR )(...)( for some positive integer n n1 1 k kn i and some division ring i kiD },...,2,1{
Theorem 2.6.10 [13] A module M is semi simple if and only if every submodule of M is a direct summand of M.
Theorem 2.6.11 [13] For a ring R the following conditions are equivalent: (a) R is semisimple if and only if all R-module are projective; (b) R is semisimple if and only if all R-module are projective;
2.7 Injective and Projective Modules Injective and projective modules or, more generally, injective and projective objects in a category, play a very important role in the development of algebra. As a tool for the investigation of injective and projective modules, we need essential and superfluous submodules as well as complements. These concepts are essentially needed also in other respects, e.g., with respect to the Radical and the Socle.
Definition A right R-module M is called an injective module if for any right R-modules A and B, any monomorphism : BAf and any homomorphism MA ,: there exists a
homomorphism : MB such that f .
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For example, and /are injective -module. On the other hand is not an injective -module because the homomorphism f :2 given by the rule )2( nnf cannot be extended to a homomorphism from to . Every nonzero proper submodule of p is quasi-injective. Also any simple module M quasi-injective since the only submodules are {0} and M. Of course, simple submodules need not always be injective (unless R is so called V-ring). Over a semisimple ring, all modules are injective and consequently quasi- injective. A semisimple R-module M is always quasi-injective. The following theorem and proposition gives some characterizations of injectivity:
Theorem 2.7.1 [35] Let E be a right R-module. Then the following conditions are equivalent: (a) E is an injective R-module; (b) Every monomorphism : BE splits, i. e., Im( ) is a direct summand of B. (c) For every monomorphism : BA of right R-module and any homomorphism, : EA , there exists a homomorphism : EB such that .
(d) Hom R (. , E) is an exact functor, i. e., for every exact sequence 0 NML 0
of right R-modules, R R R ELHomEMHomENHom 0),(),(),(0 is an exact sequence of abelian groups; (e) E has no proper essential extensions;
Theorem 2.7.2[35] For a right R-module P the following conditions are equivalent: (a) P is injective R-module; (b) P is R-injective; (c) For every right ideal I of R every homomorphism, PIh ,: there exists Py with ayah )()( for all Ia . By the works of Baer every right R-module M can be approximated by an injective module and this approximation is the injective hull E(M) of M.
38
Definition
Let i IiA }:{ be a collection of right R-modules. For each Ii , let : AAf iii 1 be an R-homomorphism. Then a sequence
f1 f2 f31 fn1 fn fn1 1 2 AAA 3 ... n AA n1 ... is called an exact sequence at An if n1 fKerf n )()Im( . The sequence is called an exact sequence if it is exact at each An . An special exact sequence of the form 0 f g CBA 0 is called a short exact sequence.
Remarks. (a) If the sequence 0 f g CBA 0 is exact, then f is a monomorphism, g is an epimorphism and gKerf )()Im( . (b) Let x RModM , consider the inclusion map : MX defined by )( xx for any Xx . Then the sequence
0 XMMX 0/ is exact, where is the canonical map.
Theorem 2.7.3 [2] Let M be any right R-module. The the following statements are equivalent: (a) M is injective; (b) Any exact sequence of the form 0 BAM 0 splits;
Definition A right R-module M is called a projective module if for any right R-module B and C, any epimorphism : CBg and any homomorphism : CM , there exists a
Homomorphism : BM such that go .
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Every free module is projective, but the converse is not true. Consider the ring R /6, which can be decomposed by R )3()2( . The ideals )2( and )3( are projective modules
but they are not free. For any n , n=/nis a quasi projective but not -projective.
The -modules /and p are not quasi-projective.
Recall:
(i) ggg is projective. (ii) fif is injective.
Proposition 2.7.4 [2] Let M be any right R-module. Then the following statements are equivalent: (a) M is projective, (b) Any exact sequence of the form 0 MYX 0 splits,
The following theorem gives some characterization of projective modules.
Theorem 2.7.5 [35] Let P be a right R-module Then the following conditions are equivalent: (a) P is projective R-module;
(b) R PHom ,.)( is an exact functor , i.e. for every exact sequence 0 NML 0 of right R-module;
R R R NPHomMPHomLPHom 0),(),(),(0
40 is an exact sequence of abelian groups; (c) P is isomorphic to a direct summand of a free module; (d) Every exact sequence 0 PML 0 of right R-module splits. It gives in [18], that if P is M-projective and the sequence 0 MMM 0 is exact, then P is projective with respect to both M and M .
Proposition 2.7.6 [2] Let M be a right module over a ring R. Then M is injective if and only if for every right ideal I of R and every R MIHomf ),( , there exists aM such that f(r)=ar for all Ir .
Proposition 2.7.7 [17] The following properties of a module M are equivalent: (a) M is projective; (b) M is a direct summand of a free module; (c) Every exact sequence 0 PML 0 splits.
Definition An element Rc is called right regular (resp. left regular) if for any r R , rcr 00 (resp. rrc ).00 If 0 rccr then c is called a regular element. For example, every non-zero element of an integral domain is regular and if F is a field, then any element of the set n FM )( is regular if and only if its determinant values is zero. Elements which are regular on one side need not be regular.
Proposition 2.7.8 [2] The following properties of a ring R are equivalent: (a) R is regular; (b) Every principal right ideal of R is generated by an idempotent element; (c) Every finitely generated right ideal of R is generated by an idempotent element; (d) Every left R-module.
41
CHAPTER III UNISERIAL AND SERIAL RINGS
Overview
A ring R is called (uni)serial if both modules RR and RR are (uni)serial. Thus uniserial rings are similar to local serial rings. Serial rings provide the best illustration of the relationship between the structure of rings and its categories of modules. There were one of the earliest examples of ring of the finite module or representation type which was introduced by Tadasi Nakayama (1912-1964) about 50 years ago. A serial ring (generalized uniserial ring ) is the terminology of Nakayama is one whose left and right free modules are the direct sum of the modules with unique composition series. Let R be a ring each of whose finitely generated left module is a direct sum of uniserial modules. Then R is serial and thus every left or right R-modules is a direct sum of uniserial modules, that is the generalized left uniserial rings of Griffith are the same as the serial rings.
3.1 Serial and uniserial rings
Definition A ring R is right serial if it is serial as a right R-module and R is serial if R is left and right serial. A ring R is called generalized uniserial if every indecomposable projective module is uniserial.
Examples The quotient ring for any integer n>1 is always serial, and is uniserial when n is a prime power. In Mathematics, a prime power is a positive integer power of a single prime number. For example: 5 = 51, 9 = 32 and 16 = 24 are prime powers, while 6 = 2 × 3, 15 = 3 × 5 and 36 = 62 = 22 × 32 are not. Semisimple Artinian rings and rings of triangular matrices over a field are serial rings.
Definition A ring R has the right Restricted Minimum Condition (RMC) if R/K is an Artinian right R- module for each essential right ideal of R.
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Following Armendariz [34], a ring R has DCC on essential right ideals if and only if
R/Soc( RR ) is right Artinian. A ring with DCC on right ideals satisfies the right RMC but the converse is not true. For example, the ring of integers has RMC but does not satisfy the DCC on essential right ideals. The following well known theorems are very useful in our investigations.
Theorem 3.1.1 If R is a right and left Noetherian right serial ring then R has the right RMC.
Theorem 3.1.2 If R is a right nonsingular right serial ring then R is right semi hereditary.
Theorem 3.1.3 If R is a right and left Noetherian right serial ring then R is a direct sum of Artinian rings and prime rings.
Theorem 3.1.4 [12] Let R be a right Artinian ring. Then the following conditions are equivalent: (a) R is generalized uniserial; Every indecomposable right R-module is quasi-projective; Every indecomposable right R-module is quasi-injective; Every indecomposable quasi-injective right R-module is quasi-projective; Every indecomposable quasi-projective right R-module is quasi-injective;
Theorem 3.1.5 [12] The following statements about a left Artinian ring R are equivalent: (a) R is generalized uniserial; (b) Every indecomposable left R-module is both quasi-projective and quasi-injective; (c) Every indecomposable left R-module is uniserial; (d) For each primitive idempotent e in R, Re and E(T(Re)) are uniserial;
Theorem 3.1.6 [35] For a right R-module M the following conditions are equivalent: (a) M is uniserial;
43
(b) The cyclic submodules of M are linearly ordered; (c) Any submodule of M has at most one maximal submodule; (d) For any finitely generated submodule KRadKMK )(/,0 is simple; (e) For every factor module L of M LSoc )( is simple or zero;
Proof ba )()( is obvious, because every R-module is uniserial. ab )()( . Let K, L be submodules of M with LK and KL . Choosing LKx ,\ kLy ,\ then we have, by (b) RyRx or RxRy . In the first case we conclude LRyx and in the second case KRxy . Both are contradictions. ca )()( and eda )()()( are obvious as Submodules and factor modules of M are again uniserial. bd ).()( Let us assume that we can find two cyclic submodules , NLK with LK and KL . Then LKLLKKLKLK )/()/()/()( and the factor of LKLK )/()( by its radical contains at least two simple summands. Therefore, the factor of LK )( by its radical also contains at least two simple summands. This contradicts d).( de ).()( We show that every non-zero finitely generated submodule NK contains only one maximal submodule: If , 21 KVV are different maximal submodules, then
21 1 //)/( VKVKVVK 2 is contained in the socle of VVN 21 )./( This is a contradiction to e).(
Theorem 3.1.7 [35] Let N be uniserial, M is a right R-module and MEndS ).( Then the following conditions are equivalent:
(a) If M is self-projective, then S NMHom ),( is uniserial;
(b) If M is self-injective, then NMHom ),(S is uniserial;
(c) If N is finitely generated and M is weakly M-injective then NMHom ),(S is uniserial;
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Proof (a) For NMHomgf ),(, it is to show that Sgf or Sfg . Let us assume fg ),Im()Im( then the following diagram can be completed commutatively for some sfgeiSs ..,.,
M
g M f f 0)Im(
ac ),()( since M is weekly M- projective and hence uniserial. (b) Now consider NMHomgf ),(, and fKergKer ).()( By factorizing suitably we obtain the commutative diagram as follows.
)(/0 f MgKerN
M and we can find some Ss with gsf . (c) It is obtained by the same proof as (b).
Definition Let M be an R-module. We say that an R-module N is subgenerated by M, or that M is a subgenerator for N, if N is isomorphic to a submodule of an M-generated module. A subcategory C of right R-module is subgenerated by M, or M is a subgenerator for C, if every object in C is subgenerated by M. We denote by M ][ the full subcategory of right R-module whose objects are all R-modules subgenerated by M.
Corollary 3.1.8 [35] (a) If M is uniserial and self-projective, then MEnd )( is left uniserial; (b) If M is uniserial and self-injective, then MEnd )( is right uniserial;
Proposition 3.1.9 [35] Let M be a left serial ring, then the following conditions are equivalent:
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For every ideal RI , / RI is a left serial ring; For idempotent Re , Re is serial and End(Re) Re is left serial;
If R R is absolutely pure, then R is also right serial; If R is a serial ring and I R is an ideal, then / IR is serial;
Proof
Two-sided ideals are fully invariant submodules of R R and the assertion follows that / RI is a left serial ring. If MI ][ is finitely generated and M-projective, then I is semiperfect in M ][ and every direct summand of I is serial. That is, if Re is finitely generated and self-projective, End(Re) is left serial. Absolutely pure is equivalent to weakly R-injective and the assertion follows R is right serial. (iv) It follows from (i), i.e., / RI is serial.
Theorem 3.1.10 [29] Let R be a semiprime serial ring which has the right RMC. Then R is Noetherian.
Proof It is easy to see that R has right Krull dimension at most 1, since R has right RMC. One can deduce that R is a Goldie ring from the fact that it is a semiprime ring of Krull- dimension at most 1.
Let 1 ... n ReReR where ei }{ is a set of mutually orthogonal primitive idempotent of R.
Suppose that i nkkiReSoc )(,...,2,1,0)( and j nkjReSoc ,,...,1,0)( since R has right RMC, we have i Re is Artinian for each ki .,...,2,1 Set 1 ... k ReReA and
k 1 n ReReB ,... then A is clearly a two sided ideal of R with AR is Artinian of R and
R BAR .We shall show that AB is also a two sided ideal of R. It is clear that BA .0
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Therefore, 2 BBAAAB ,0)()( since R is semiprime. It follows that AB=0. This shows that B is also a left ideal of R, i.e., B is a two sided ideal . Thus R has a ring decomposition
BAR with A is a semiprime Artinian and BSoc B ).( Therefore, splitting of the socle we may assume rSoc .0)( The restricted minimum condition implies that r .0)( So R is semi hereditary by [25]. Let 21 ... REE be an ascending chain of finitely generated essential right ideals. Choose regular in
1 * * E1. Then we have a chain of right R-module so that 1 EERc 2 ...)()( R so that
* * 1 2 RccEcER ....)()( This chain of left ideals stops by the restricted minimum
* * * * condition. Therefore, n n1 cEcE . Thus EE nn 1 and hence EE nn 1 by taking stars again and using the fact that the E ’s are projective. Hence R is Noetherian. i
Theorem 3.1.11 [29] Let M be a serial ring which has the right and left RMC, Then R is Noetherian.
Proof As above, we see that R has right and left Krull dimension at most 1. Let N be the prime radical of R. Then N is nilpotent by [23]. Consider the ring NRR ./ Then R is semiprime and has right and left RMC, hence R is Noetherian. Applying a result of [24] R has a (ring) decomposition BSR , where S is an Artinian ring and B is a direct sum of hereditary prime rings. In particular, BSoc B 0)( by [25].
Assume that / NSS where S is a two sided ideal of R. In order to end the proof it is enough to show that S is Artinian and there are idempotents e and f of R such that
RfeRS . By the above arguments SRSoc R ,0)/( since R has right RMC. It follows that there is an idempotent e of R such that eRS . Using the same argument, we can find an idempotent f of R such that RfS .Thus we have RR RSR with SRNN ,/ since R s a semiprime serial ring having the right and left RMC and hence R is Noetherian. For the ring S, since / NS is semiprime Artinian, it impies that SJN )(/ where SJ )( is the Jacobson radical of the ring S. Thus SJ )( is nilpotent and SJS )(/ is
47
Artinan. From this we conclude that S is a semiprimary ring. It follows that S is an Artinian ring, since S has right and left RMC. Therefore, S is a Noetherian ring.
Lemma 3.1.12 [21] Let R be a ring. Then the following conditions are equivalent: There is a uniserial right module A over R that is not quasi-small;
There is a uniserial right module A over R such that AA em ;
There is a finitely generated uniserial right module A satisfying AA em ;
Theorem 3.1.13 [30] Let R be an Artinian ring such that / JR is a direct product of matrix rings over finite dimensional division rings. Then the following are equivalent: (a) Every indecomposable injective left R-module is uniserial; (b) is right serial.
Theorem 3.1.14 [35] Let R be a right perfect ring. Then the following conditions are equivalent: (a) R is a serial Artinian ring; (b) Every cyclic module is a direct sum of an injective module and a uniform small module; (c) Every local module is either injective or uniform small;
Theorem 3.1.15 [35] Let M be a non-zero uniserial R-module. Then the following conditions are equivalent: (a) Submodules and factor modules of M are again uniserial; (b) M is uniform and finitely generated submodules of M are cyclic. (c) )( MMRad if and only if M is finitely generated, MSoc 0)( if and only if M is finitely cogenerated; (d) If M is Noetherian, there exists a possibly finite descending chain of submodules M
... MMM 21 ... with simple factors MM ii 1 ;/ (e) If M is Artinian, there exists a possibly finite ascending chain of submodules
0 SSS 210 ... with simple factors 1 SS ii ;/
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(f) If M as finite length, then there is a unique composition series in M;
Proof (a) It is obvious since submodules and factors modules are uniserial. (b) It is evident that every submodule is essential in M.
Assume RkRkK 21 ..... r NRk . Since the Rki ’s are linearly ordered, we have
RkK j for some j r. (c) Here )( MMRad implies )( MMRad and MSoc 0)( means )( MMSoc
(d) For 1 MRadM ),( / MM 1 is simple, since all submodules of M are finitely generated. Now we define recursively i1 MMRadM ii .)(
(e) Since M is Artinian, put 0 MSocS .0)( All factor modules of M are Artinian and we construct Si1 by 1 ii SMSocSS i )./(/ (f) M is Noetherian and Artinian and the series in (d) and (e) are finite and equal to each other.
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CHAPTER IV UNISERIAL AND SERIAL MODULES
Overview Uniserial and serial modules appear in many contexts. By an adaptation of basic properties of uniserial and serial rings over arbitrary ring, we have created a few structures of uniserial and serial modules over an endomorphism rings. Right uniserial rings can be referred to as right chain ring (Faith 1999) or right valuation ring. By the same token uniserial modules have been called chain modules and serial modules semichain modules. In 1930s Gottfried Kothe and Keizo Assano introduced the term “Einreihig’’(literally one- series) during investigation of ring over which all modules are direct sum of cyclic submodules. For this reason uniserial is introduced to mean “Artinian principle ideal ring” even as recently as the 1970s. Expanding on Kothe’s work Tadashi Nakayama used the term generalized uniserial to refer to an Artinian serial ring. Nakayama showed that all modules over such rings are serial. Warfield used the term homogeneously serial module for a serial module with the additional property that for any two finitely generated submodules A and B such that BJBAJA ),(/)(/ where J denotes the Jacobson radical of the modules. In this chapter, we investigate the properties of uniserial and serial modules over associative endomorphism ring as a generalization of uniserial and serial rings over associative arbitrary rings.
4.1 Uniserial and serial modules
Definition A right R-module M is called uniserial if the lattice of its submodules is linearly ordered by inclusion, i.e., if X and Y are submodules of M, then s YX or s XY . As an example, we see that 4 is a uniserial -module. A ring R is said to be right (resp. left) uniserial if it is uniserial as a right (resp. left) R-module. A ring R is called a local ring if it has a unique maximal right (or left) ideal. Left and right uniserial rings are in particular, local rings.
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A right R-module is called a serial module if it is a direct sum of uniserial modules. It is to be noted that submodules and factor modules of serial modules need not be serial. A ring R is said to be right (resp. left) serial if it is serial as a right (resp. left) R-module. So a ring R is right serial if there are orthogonal idempotents ,,, eee of R such that and each 21 n
i Re is uniserial as a right R-module. We say R a serial module if it is a left and right serial module. Note that every uniserial module is serial but serial modules need not be uniserial.
Examples
(i) Dn ={all divisors of n} i.e. D12 ={1, 2, 3, 4, 6, 12} (ii) In Boolean Algebra the set of all subsets of are ordered by inclusion. (iii) Every Simple module is uniserial. (iv) Every function having finite length is uniserial. F[x,y]={x3, x2y, y3} is uniserial since its length is 3. (v) Every semisimple module is serial’
(vi) 4 is uniserial module. n n n 2 n n-1 n n n n (vii) P =1/p p/p p /p … p /p p /p =0, here P is uniserial. The following Proposition gives some characterizations of a serial right R-module which implies R MEndS )( to be right or left serial.
Proposition 4.1.1 [35] Let M be a serial right R-module.
(1) Suppose that X is an R RMEnd )),(( -submodule of M. Then X and M/X are serial modules.
(2) If M is finitely generated and self-projective, then R MEnd )( is right serial.
(3) If M is finitely generated and weakly M-injective, then R MEnd )( is left serial.
Definition A subset L of a left R-module M is called a generating set of M if MRL . We also say L generates M or M is generated by L. If there is a finite generating set in M, then M is called
51 finitely generated. For example, every ring is generated by its unit and the left principal ideals are just the cyclic submodules of R R.
Note A finitely generated module over a field is simply a finite dimensional vector space and a finitely generated module over the integers is simply a finitely generated abelian group.
Theorem 4.1.2 [35] Let M be an Artinian right R-module which is a projective generator in M ][ . Then the following statements are equivalent: (a) M is serial and every indecomposable quasi-injective module in M ][ is uniserial; (b) Every finitely generated indecomposable module in M ][ is uniserial;
Corollary 4.1.3 [35] Let R be an Artinian right R-module. Then the following statements are equivalent: (1) R is serial and every indecomposable quasi-injective right R-module is uniserial; (2) Every finitely generated right R-module is serial; (3) Every right R-module is serial; (4) Every finitely generated indecomposable right R-module is uniserial;
Proposition 4.1.4 [35] Let N be a serial right R-module M R -module and S=End(M). Assume N is finitely generated. Then the following conditions are equivalent:
(a) If M is self-projective, then S R NMHom ),( is a serial S-module;
(b) If M is weakly M-injective, then R NMHom ),( S is a serial S-module; (c) If MN ][ is finitely generated and M-projective, then N is semiperfect in M ][ and every direct summand of N is serial.
Definition
Let M be a finitely generated R-module and MRadM )(/ is semisimple and M1 be a simple R-module. By M )( we denote the number of simple summands in a decomposition of MRadM )(/ and by MM 1 ),( the number of summands isomorphic to
52
M1 in a decomposition of MRadM ).(/ Then M )( is exactly the length of MRadM )(/ and MM 1 ),( the length of the M1-generated, homogeneous component of MRadM )(/ .
Theorem 4.1.5 [35] Let M is a finitely generated R-module and MRadM )(/ is semisimple. Then the following conditions are equivalent:
(a) If LKM , then for every simple R-module M1, 1 1 MLMKMM 1 ),(),(),( . (b) If M is self-projective, then a finitely M-generated module K is M-cyclic if and only if
1 MMMK 1 ),(),( for every simple module E. (c) If M is serial, then, for every finitely M-generated submodule MK , MK )()(
The following Theorem gives some properties that can imply M to be serial.
Theorem 4.1.6 [35] Let M be an Artinian right R-module which is a projective generator in M ][ . Then the following statements are equivalent: (a) M is serial and every indecomposable quasi-injective module in M ][ is uniserial; (b) Every finitely generated indecomposable module in M ][ is uniserial;
Corollary 4.1.7 [35] Let R be an Artinian right R-module. Then the following statements are equivalent: (1) R is serial and every indecomposable quasi-injective right R-module is uniserial; (2) Every finitely generated right R-module is serial; (3) Every right R-module is serial; (4) Every finitely generated indecomposable right R-module is uniserial.
Theorem 4.1.8 [35] For a finitely generated R-module M the followings are equivalent: Every finitely M-generated module is serial;
Every factor module of MM is serial; M is serial, and every indecomposable injective (uniform) module in M ][ is uniserial;
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M is serial, and every module, which is finitely generated by cyclic, uniserial modules in M ][ is serial. If there is a generating set of cyclic uniserial modules in M ][ , then (a) to (d) are also equivalent to: Every finitely generated module in M ][ is serial;
Proof cb ).()( Let be an indecomposable, injective module in M ].[ Then is M-generated and every finitely generated submodule of is contained in a finitely M-generated submodule of. Since is uniform, we obtain from (b) that any sum of two M-cyclic submodules of is uniserial. Hence every finitely M-generated submodule of (and also itself) is uniserial. dc ).()( If c)( holds, then in particular the injective hulls of the uniserial modules in M ][ are uniserial. ed ).()( Given a generating set as demanded, every finitely generated module in M ][ is a finite sum of cyclic uniserial modules.
Theorem 4.1.9 [35]
Let M be an Artinian module and R MEndS ).( Then the following conditions are equivalent: (a) If M is semi-projective, then S/J(S) is left semisimple and J(S) is nilpotent; (b) If M is finitely generated and self-projective, then S is left Artinian and M satisfies the ascending chain condition(ACC) for M-generated submodules; (c) If M is self-injective, then S is right Noetherian; (d) If M is self-injective and self-projective, then S is right Artinian;
Proof (a) If the ring R to satisfy DCC for cyclic left ideals, then RJR )(/ is left semisimple and RJ )( is left nilpotent. Thus we obtain that SJS )(/ is left semisimple and SJ )( is left nilpotent. For SJJ )( the descending chain of R-submodules 2 MJMJMJ 3 ...
54 has to become stationary after finitely many steps. Hence we get, for some n and JB n that MBMB 2 . Assume J not to be nilpotent. Then this is also true for B and hence there exists Bc with cB .0 Let Mc denote a minimal element in the set cBBcMcM }.0,|{ Since 0 MBc MBBc, there exists BBcd with Bd 0and McMBcMd . By minimality of Mc, this means Md = Mc and hence Mbc = Mc for some Bb .
M being semi-projective, there exists Sf with fbc=c, i.e., cbc for JBbfb .
Since b is b nilpotent, this implies c=0, which contradicts the assumption of c. Therefore J has to be nilpotent. (b) For the given assumptions, for every left ideal SI , we have MIMHomI ).,( If M is finitely generated and self-projective, then S is left Artinian. Thus we conclude that S S is Artinian.
Now every M-generated submodule of M is of the form MI with S SI . Hence S S is Noetherian. Hence if M is finitely generated and self-projective, then S is left Artinian and M satisfies the ascending chain condition(ACC) for M-generated submodules. Thus we can deduce the ascending chain condition for these submodules.
(c) We have MKeIMHom ),/( for every finitely generated right ideal SI S . Therefore the descending chain condition for submodules of type IKer )( yields the ascending chain condition for finitely generated right ideals SI S , i.e., S S is Noetherian. S is right Noetherian. (d) We have SJS )(/ is semisimple and SJ )( nilpotent. Since RJR )(/ is a left semisimple ring and RJ )( is nilpotent, hence RR is Noetherian and so S S is Noetherian. Hence S is Artinian. We assume an R-module M semi-injective if, for any factor module N of M, every diagram with exact row 0 k MN g
M
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can be extended by an : MMh with gkh . This is obviously the case if and only if, for every R gKefSgfSSMEndf R MKefMHom ),/(}0)(|{,)( . Hence S is right Artinian.
Theorem 4.1.10 [35] For a ring R the following assertions are equivalent: (a) Every finitely generated module in right R-module is serial;
(b) Every factor module of R RR is serial; (c) R is left serial, and every indecomposable injective module in right R-module is uniserial; If these properties hold, R is also right serial.
If R R is noetherian, then ca )()( are equivalent to: (d) R is (left and right) serial;
Proof The equivalence of (a), (b) and (c) are obvious. In particular, the finitely presented left modules are serial and hence R is right serial. If R is noetherian, all finitely generated R- modules are finitely presented, and hence da ).()( If the module M is hereditary in M ],[ then the factor modules of injective modules are again injective in M ][ modules in M ][ are uniserial in the following situation. Hence the theorem.
Theorem 4.1.11 [35] For an R-module M of finite length the following are equivalent: Every module in M ][ is serial; Every finitely generated module in M ][ is serial; Every finitely generated indecomposable module in M ][ is uniserial; Every non-zero finitely generated module N in M ][ contains a non-zero N-projective (and N-injective) direct summand; Every finitely generated indecomposable module in M ][ is self-injective and self-projective;
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Theorem 4.1.12 [35] For a ring R the following assertions are equivalent: Every module M in right R-module is serial; R is a serial ring and left (and right) Artinian; Every module M in left R-module is serial;
Proof ba )()( is clear, since R is left and right serial. Also every right R-module is a direct sum of indecomposable modules. Hence R is left pure semisimple and in particular, left Artinian. Now we see that the lengths of the indecomposable, i.e. uniserial modules in right R-module are bounded (by the length of R R ). Therefore R R and RR are of finite type and RR is Artinian. ab ).()( If R is a serial left Artinian (hence left Noetherian) ring, then we have the finitely generated modules in right R-module are serial. Hence it follows that all modules in right R-module are serial. cb )()( is obtained symmetrically to ba ).()(
The next result characterizes serial rings in terms of left module and also serves to describe their left and right modules.
Theorem 4.1.13 [2] If R is a left Artinian ring then the following conditions are equivalent: (a) R is a serial ring; Every left R-module is a direct sum of uniserial modules; Every finitely generated indecomposable left R-module is uniserial;
Lemma 4.1.14 [2] Let M be a right R-module. Then for M ,0 the following conditions are equivalent: (a) M is uniserial; (b) M has unique composition series; (c) The radical series JMM ... l MJ 0 M is a composition series for M;
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(d) The socle series 2 MSocMSoc l )(...)()(0 MMSoc is a composition series for M;
Theorem 4.1.15 [35] Let A be a sub ring of a ring R. Suppose that A is Noetherian and R is finitely generated as A-submodule. Then R is Noetherian ring.
Proof Since R is finitely generated as A-module hence all ideals of are also A-submodule of R. Since A-submodule satisfy the ACC, so the ideals of R satisfy the ACC. Hence R is Noetherian ring.
Definition Let M be a right R-module. An injective module E is called an injective hull of M if there is
* a monomorphism : EMf such that )( EMf . Equivalently, an injective hull E(M) for a module M is any injective module which is an essential extension of M. By Zorn’s lemma, injective hull always exists and is unique up to isomorphism. For example, we can take as an injective hull for . A right R-module is quasi-injective if and only if M is fully invariant submodule of E(M). We use the following facts: (i) Every right R-module can be embedded in an injective module.
(ii) Property C1: If MX and M is injective, them there is a direct summand Y of
* M such that MYX
* A submodule MX is called closed in M if YXMYX
A projective module P is called projective cover of MR if there exists an epimorphism
: MP such that Ker )( P .
Note Every module has a projective cover. A ring R is said to be right perfect if every right R-module has a projective cover. A right R-module M is called semi perfect if every homomorphic image of M has projective cover.
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Definition A module is called uniform if the intersection of any two non-zero submodule is non-zero. Uniform modules are uniserial.
Example
(i) 6 is uniform module.
Theorem 4.1.16 [2] An Artinian ring R is serial if each of its finitely generated indecomposable module is local.
The following lemma is an extension of the above theorem
Lemma 4.1.17 Let M be a right R-module and E(M) be its injective hull. Then M is uniform if and only if E(M) is indecomposable.
Proof
( ) Assume that M is uniform and E(M) = MM 21 . Since M is an essential submodule of E(M), we have MM 1 0 and MM 2 .0 By assumption, we have
1 MMMM 2 ,0)()( a contradiction. Thus, E(M) is indecomposable. ( ) Assume that E(M) is indecomposable. Let X, Y be submodules of M and E(X), E(Y) be injective hulls of X and Y respectively. Suppose that YX .0 Then YEXE .0)()( Since MEXEO )()( is exact and E(X) is injective, E(X) is a summand of E(M). Therefore, by assumption, E(X) = 0 and E(X) = E(M). Thus, X = 0 or Y = 0 proving that M is uniform.
Lemma 4.1.18 [26] Any uniform right R-module is uniserial.
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The following lemma is an extension of the above lemma.
Lemma 4.1.19 Let M be a right R-module. Then M is uniserial if and only if every factor module of M is uniform.
Proof ( ) Assume that M is uniserial. Then every factor module of M is uniserial. Thus, every factor module of M is uniform. ( ) Assume that every factor module of M is uniform. Let A, B be submodules of M. Since M is uniform, BA .0 Suppose that BA and AB . Then BAA 0)/( and BAB .0)/( But we know that BAA )/( BAB .0/( This is a contradiction with the property of the uniform module BAM )./( It follows that A B and AB . Thus, M is uniserial.
Lemma 4.1.20 [35]
Let 1 ,..., MM K be a self-projective R-module with all Mi’s cyclic and uniserial, then M/K is serial.
This is an extension of the above lemma.
Lemma 4.1.21 Every finitely generated uniserial module is cyclic.
Proof Let U be a finitely generated uniserial right R-module. Then U contains a maximal submodule V, say. Let VUx . Then VxR . Since U is uniserial, we must have xRV . It follows that xRU . This completes the proof.
Theorem 4.1.22 Let M be an Artinian right R-module which is a projective generator in MNM ]],[ and R NMHomf ).,( If M is serial and every indecomposable quasi-injective module in M ][ is uniserial, then Mf )( is serial.
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Proof Since M is Artinian projective generator in M ],[ M has finite length. By the property of
n serial modules, we have MM i where each M i is uniserial. It follows that each Mi is i cyclic and thereforeMf i )( is cyclic and uniserial.
n n From )( MfMf i Mf i )( and the fact that the injective hull of Mf i )( is i1 i1
n uniserial, so we can see that Mf i )( is a serial module. Therefore Mf )( is serial. i1
Theorem 4.1.23 Let M be an Artinian right R-module which is a projective generator in M ].[ If every factor module of local in M ][ is quasi-injective, then M is serial.
Proof
n
Since M is projective in M ][ and Artinian, we have MM i where each M i is a i local module. By assumption, every factor module of Mi is quasiinjective. Then we can see that every factor module of Mi is uniform. That is each Mi is uniserial, proving that M is serial. In general, it is well-known that every epimorphic image of a serial module needs not to be serial. The following Theorem gives some conditions for an epimorphic image of serial module to be serial.
Theorem 4.1.24 [3] For a right Artinian and right self injective ring R, the following conditions are equivalent: (a) R is right serial; (b) R is Artinian and serial;
The following theorem is an extension of the above theorem.
Theorem 4.1.25 Let M be an Artinian right R-module which is a projective generator in M ].[ Then the following statements are equivalent:
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(a) Every module in M ][ is serial; (b) Every finitely generated module in M ][ is serial;
Proof ab ).()( Since M is a generator in M ],[ every finitely generated module in M ][ is finitely M-generated. That is, every finitely generated module in M ][ is serial. ba ).()( Since M is an Artinian projective generator in M ],[ M has finite length. Then we have every module in M ][ is serial.
Theorem 4.1.26 [35] For a ring R, the following conditions are equivalent:
(a) RR is right Artinian;
(b) RR is right Noetherian. (c) Every finitely generated right R-module is finitely cogenerated;
Theorem 4.1.27 Let M be a R-module over a left serial ring R, then the following assertions hold:
(a) Every left R-module in R M is serial;
(b) Every right R-module in M R is serial; (c) Every finitely generated module is serial;
Proof (a) If R is a serial left Artinian (hence left Noetherian) ring, then, the finitely generated modules in left R-module is serial. Hence it follows that all modules in R M are serial. (b) If M is a serial right artinian (hence left Noetherian) ring, then, the finitely generated modules in right R-module is serial. Hence it follows that all modules in M R are serial. (c) R is left and right serial. Since every module in M is a direct sum of indecomposable modules, M is left pure semisimple and hence every finitely generated module is serial.
Theorem 4.1.28 [26] Let R be a ring each of whose finitely generated left modules is a direct sum of uniserίal
62 modules. Then R is serial and thus every left or right R-module is a direct sum of uniserial modules.
The following theorem is the generalization of the above theorem.
Theorem 4.1.29 Let M be an Artinian right R-module which is quasi-p-injective and a projective subgenerator with MEndS R )( . Then the following conditions are equivalent: (a) M is serial; (b) S is Artinian and serial;
Proof ba )()( . Since M is serial, finitely generated and quasi-projective, the endomorphism ring MEndS R )( is right serial and hence left serial. This proves that M is serial. ab )()( . We know that every indecomposable left (right) S-module is uniserial. Using the equivalence between M ][ and S-module we can see that every indecomposable
n module in M ][ is uniserial. Since MM i with each M i is indecomposable, we i1 conclude that M is a serial right R-module.
Definition An idempotent element of a ring R is an element a such that 2 aa .That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that . 32 ... aaaa n for any positive integer n. We observe that if there are m factors, there will be 2m idempotents. We can check this for
2 the integers mod 6. Let R=6 . Since 6 has 2 factors (2 and 3) it should have 2 idempotent. 2 000 (mod 6) 12 1 1(mod 6) 22 4 4 (mod 6) 2 393 (mod 6) 2 4164 (mod 6)
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2 1255 (mod 6) From these computations, 0, 1, 3 and 4 are idempotents of this ring, while 2 and 5 are not.
Note (i) A trivial idempotent refers to either of the elements 0 and 1, which are always idempotent. (ii) A primitive idempotent is an idempotent a such that aR is directly indecomposable. (iii) Two idempotents a and b are called orthogonal if ab = ba = 0. If a is idempotent in the ring R (with unity), then so is b = 1 – a. Moreover, a and b are orthogonal.
Lemma 4.1.30 [18] For a ring R, the following conditions are equivalent: (1) R is semisimple; (2) R is semiprime and right Artinian; (3) R is semiprime and satisfies the descending chain condition on principal right ideals.
As a generalization of this Lemma, we investigate the following theorem.
Theorem 4.1.31 Let M be a self-projective finitely generated right R-module which is a self-generator. Then the following conditions are equivalent: (1) M is semisimple; (2) M is semiprime Artinian; (3) M is semiprime and satisfies the descending chain condition on M-cyclic submodules.
Proof )2()1( and )3()2( are obvious. We prove ).1()3( By Theorem 2.9[14], if M is a semiprime module, then R MEndS )( is a semiprime ring. Since M is finitely generated, self-projective and a self-generator, it follows from [21, 43.10] that S satisfies descending chain condition on principal right ideals. Hence by above Lemma [4.1.38], the ring S is semisimple. For orthogonal primitive idempotents 21 ,,, eee n in S, then we get
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1 21 eee n .
This implies that 1 2 n )()()( MMMeMeMeM 21 M n with
ii MeM )( indecomposable. Since M is a self-generator, there is a nonzero element f in S satisfying MMf 1.)( Since S is a regular ring, there is a nonzero element g in S such that fgff and hence fg is an idempotent in S. It follows that Mfg )( is a direct summand of
M. Let MfgN ).( Then there exists a submodule P of M such that PNM . By
Modular Law, we get 11 1 PMNPNMM )()( proving that N is a direct summand of M1. But M1 is indecomposable and so it is simple. Applying similar arguments, we can show that 32 ,,, MMM n are simple. Therefore, M is a semisimple right R-module.
Lemma 4.1.32 [35] Let M be a self-injective module and let X be a fully invariant submodule of M. Then X is self-injective.
Lemma 4.1.33 [35]
Let M be an indecomposable self-injective module and let R MEndS )( its endomorphism ring. Then S is local.
By using above lemma we have investigated the following theorem.
Theorem 4.1.34 Let M be a Noetherian uniserial right R-module. Then every submodule of M is fully invariant in M and if M is self-injective then every submodule X of M is self-injective and XEnd )( is a local ring.
Proof Let : MMf be an endomorphism and f .0 Suppose that there exists a submodule
X 1 of M for which )( XXXf 121 . Since XfKer 1 ,)( we have 2 )( XXXf 23 .
Proceeding in this way, we get a strongly ascending chain XXX 321 of
65 submodules of M, a contradiction since M is Noetherian. Thus, every submodule of M is fully invariant. By Lemma 4.1.40 and Lemma 4.1.41, XEnd )( is a local ring.
Theorem 4.1.35
Let M be a right R-module and R MEndS .)( If M is finitely generated, Artinian and quasi-projective, then S S is right Artinian.
Proof For every right ideal SI , we have IMMHomI .),( Let
n IIII 123 be a descending chain of right ideals of S. Then we have
n MI 3 2 1 MIMIMI )()()()( is a descending chain of M. By assumption, this chain becomes stationary after finite number of steps. Then k k ),( ik ik MIMHomIMIMHomI ),,( for Ni . Hence,
S S is right Artinian.
Theorem 4.1.36
Let M be a right R-module and R MEndS .)( If M is Artinian and quasi-injective, then
S S is left Noetherian.
Proof Since M is quasi-injective, for every finitely generated left ideal I of S,
MIKerMHomI ).),(/( Let III 321 be an ascending chain of finitely generated left ideal of S. Then IKer n )( 3 2 IKerIKerIKer 1 .)()()( Since
M is Artinian, there exist k such that k IKerIKer ik )()( for positive integer i. It follows that k k ik IMIKerMHomMIKerMHomI ik .)),(/()),(/( Therefore, S S is left Noetherian.
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CONCLUSION
In this study, we developed some structures of uniserial and serial modules over associative endomorphism rings by modifying some structures of uniserial and serial rings over associative arbitrary rings. As generalizations of uniserial and serial rings and modules over associative arbitrary rings, some characterizations of Noetherian and Artinian uniserial and serial modules over associative endomorphism rings are investigated in the present study.
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