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I a STUDY on UNISERIAL and SERIAL RINGS and MODULES By i 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 iii 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 iv DEDICATED TO MY PARENTS AND TEACHERS v 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. vi 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. vii 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 1 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. 3 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. 4 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.
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