Luận án tiến sĩ: Injectivity, Continuity và CS Conditions trên Group Rings

INJECTIVITY, CONTINUITY, AND CS CONDITIONS ON GROUP RINGS A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Adel Naif M. Alahmadi November 2006 UMI Number: 3240004 Copyright 2006 by Alahmadi, Adel Naif M. All rights reserved. UMI Microform 3240004 Copyright 2007 by ProQuest Information and Learning Company.

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Alahmadi All Rights Reserved This dissertation entitled INJECTIVITY, CONTINUITY, AND CS CONDITIONS ON GROUP RINGS by ADEL NAIF M. ALAHMADI has been approved for the Department of Mathematics and the College of Arts and Sciences by S. Jain Distinguished Professor of Mathematics Benjamin M. Ogles Dean, College of Arts and Sciences ALAHMADI, ADEL NAIF M., November 2006, Mathematics INJECTIVITY, CONTINUITY, AND CS CONDITIONS ON GROUP RINGS (81pp.) Director of Dissertation: S.

Jain Almost self-injective, continuous, quasi-continuous (also known as π-injective), and CS modules are generalizations of injective modules. The main aim of this dissertation is to study almost self-injective, continuous, quasi-continuous, and CS group rings. CS group algebras were initiated by Jain et. They showed that K[D∞ ] is a CS group algebra if and only if char(K) 6= 2.

Behn extended this result and showed that if K[G] is a prime group algebra with G polycyclic-by-finite, then K[G] is a CS-ring if and only if G is torsion-free or G ∼ = D∞ and char(K) 6= 2 [5]. As a consequence, such a group algebra K[G] is hereditary excepting possibly when K[G] is a domain. We show that if K[G] is a semiprime group algebra of polycyclic- by-finite group G and if K[G] has no direct summands that are domains, then K[G] is a CS-ring if and only if K[G] is hereditary if and only if G/∆+ (G) ∼ = D∞ and char(K) 6= 2. Furthermore, precise structure of a semiprime CS group algebra K[G] of polycyclic-by-finite group G, when K is algebraically closed, is also provided.

Among others, it is shown that (i) every almost self-injective group algebra with no nontrivial idempotents is self-injective, (ii) if G is a torsion group and the group algebra K[G] is quasi-continuous then G is a locally finite group, and (iii) for any group G, if K[G] is continuous then G is locally finite. As a consequence, it follows that a CS group algebra K[G] is continuous if and only if K[G] is principally self- injective if and only if G is locally finite. The properties of endomorphism rings of almost self-injective indecomposable modules have been investigated. It is shown that the endomorphism ring of a unis- erial almost self-injective right module is left uniserial.

For a domain D, it is proved that D is right almost self-injective if and only if D is a two sided valuation domain. Jain, Distinguished Professor of Mathematics Preface The work in this dissertation is an account of author’s original research in certain aspects of group rings, modules, and ring theory. It has been influenced by the works of a number of algebraist including Baba, Behn, Chatters, Dung, Faith, Farkas, Hajarnavis, Harada, Huynh, Jain, Kanwar, López-Permouth, Mohamed, Müller, Oshiro, Passman, Srivastava, Utumi and others. A right R-module M is said to be continuous if every complement submodule is a summand and every submodule isomorphic to a summand is itself a summand.

Modules in which complement submodules are summands are called CS modules. A module M is said to be almost self-injective if for each submodule X of M and for each R-homomorphism f : X −→ M either f can be extended to M or there exists a nonzero direct summand M1 of M and a homomorphism g : M → M1 such that g ◦ f = π ◦ iX , where π : M → M1 is a projection of M onto M1. A quasi-injective module is continuous (hence CS) and almost self-injective. A ring R is said to be right CS (resp.

continuous, almost self-injective) if the right R-module R is CS (resp. continuous, almost self-injective). The study of continuous rings dates as far back as 1937 when von Neumann introduced these rings in his study of continuous geometry [41]. Chatters and Hajarnavis introduced the concept of CS rings [9].

Continuous as well as CS rings and modules have been subjects of active research by many authors. In this dissertation we study these injectivity conditions on group rings and modules. We study the ring theoretic properties and the structure of the group G when K[G] is CS or continuous. The dissertation contains four chapters: Preliminaries, Semiprime CS Group Rings, Continuous and Almost Self-Injective Group Algebras, Almost Self-Injective Modules.

Chapter 1 introduces terminologies, notations, and basic results which contain the fundamental tools for the rest of the dissertation. In Chapter 2, characterizations for a semiprime CS group algebra K[G] of a polycyclic-by-finite group G without domains as summands are obtained. It is shown that K[G] is CS if and only if it is hereditary. Our techniques include using the twisted group algebra of the infinite dihedral group.

This enables us to obtain the precise structure of a CS semiprime group algebra of a polycyclic-by-finite group when K is algebraically closed under the assumption that the group algebra has no direct summands which are domains. It is well-known that a group algebra K[G] is self-injective if and only if G is a finite group. In Chapter 3, we consider continuous and almost self-injective group algebras. In the first section, we consider a property (P) which is stronger than almost self-injectivity.

For a ring R with no nontrivial idempotents the two concepts coincide. It is shown that if a group algebra K[G] satisfies the property (P), then it is self-injective. As a consequence, an almost self-injective group algebra with no nontrivial idempotent is self-injective. Continuous group algebras are studied in the second section.

we show that a group algebra K[G] is continuous only if G is locally finite. We also provide examples to show that the converse is not true in general. In Chapter 4, we study almost self-injective indecomposable modules and their endomorphism rings. We show that the endomorphism ring of almost self-injective indecomposable module is local.

Also, it is shown that the endomorphism ring of a uniserial almost self-injective right module is left uniserial. In the last section, we study the relationship of other injectivity conditions with almost injectivity. Dedications To my parents: Muzon and Naif To my family: my wife Nsreen, my daughters Sereen and Aljuri To all my brothers and sisters. Acknowledgements I would like to express my warmest thanks to my advisor Dr.

Jain, Dis- tinguished Professor of Mathematics and Director of the Ohio University Center for Ring Theory and its Applications, for his excellent guidance, support, and patience during my work on this dissertation. I would also thank him for awarding me two times the Distinguished Professor Scholarship in 2003-04 and 2004-05. I would like to thank Dr. Huynh and Dr.

López-Permouth for their encouragement and help whenever I needed. My warmest thanks are also extended to my friends Husain S. Al-Hazmi and Noyan F. for their collaboration and numerous mathematical discussions.

My sincere love and special thanks go to my wife Nsreen and daughters Sereen and Aljuri for their support throughout during my studies without which I would not have been able to achieve this work. Special gratitude goes to my brothers and sisters and other relatives for their prayers and best wishes. In particular, I would like to express my thanks to my sister Ameenah and her husband Ahmad Aziz and their children for their support. I extend the most sincere and special appreciation to my mother (Muzon) and my father (Naif) for their support, and belief in higher education.

I would like to acknowledge the Goernment of Saudi Arabia for providing me financial support and King Abdulaziz University for granting me academic leave for higher studies at Ohio University. Last but not least, I thank the Ohio University Center of Ring Theory and its Applications, and Department of Mathematics for providing me necessary facilities and opportunity to teach. Above all, I thank Almight Lord (Allah) for giving me the strength to complete this work and for His unlimited mercies. Contents Abstract 4 Preface 6 Dedication 9 Acknowledgements 10 1 Preliminaries 14 1.1 Definitions, Notations, and Basic Results On Modules and Rings .2 Introduction to Group Rings.

26 2 Semiprime CS Group Rings 33 2.1 Twisted Group Algebras Over Subgroups Of D∞ .2 Semiprime CS Group Rings Of Polycyclic-By-Finite Groups. 37 12 13 3 Continuous and Almost Self-Injective Group Algebras 45 3.1 Almost Self-Injective Group Algebra .2 Continuous Group Algebra. 52 4 Almost Self-Injective Modules 59 4.1 Endomorphism Rings of Almost Self-Injective Modules .2 Almost Injectivity and Other Injectivity Conditions. 70 Bibliography 77 Chapter 1 Preliminaries In this chapter we give the basic concepts and terminologies used throughout in the dissertation.

In the first section, we list definitions, notations and some basic facts on rings and modules theory. In the second section, we introduce the concept of group rings that will be needed for Chapters 2 and 3. Finally, in the last section, we state some preliminary results on group rings, rings, and modules theory.1 Definitions, Notations, and Basic Results On Modules and Rings Throughout this dissertation, unless otherwise specified, R will denote a ring with unity, and all modules are right unital. When the ring is clear from the context, 14 15 an R-homomorphism will be expressed as homomorphism.

For two right R-modules M and N , HomR (M, N ) denotes the additive group of all homomorphisms from M into N. EndR (M) denotes the ring of all endomorphisms of M. A submodule N of M is said to be an essential submodule of M, denoted by N ⊆e M, if for every nonzero submodule L of M, L ∩ N 6= 0. We say that M is an essential extension of N , if N is an essential submodule of M.

For two submodules A and B of a module M, B is said to be a complement of A in M if B is maximal with respect to the property that A ∩ B = 0. By Zorn’s Lemma, such a complement B always exists. A submodule C of M is called a complement in M if C is a complement to a submodule of M. A submodule C of M is called closed in M if it is the maximal essential extension of itself in M, that is, whenever L is a submodule of M with C ⊆e L then C = L.

Closed submodules are precisely complements in M. For modules A ⊆ C ⊆ M, C is said to be an essential closure of A in M if C is a maximal essential extension of A in M, equivalently, C is minimal among closed submodules of M containing A. An essential closure C of A in M is a closed submodule of M. A module is said to be decomposable if it is a direct sum of two nonzero sub- modules.

A nonzero module is said to be indecomposable if it is not decomposable. Clearly, a ring R is indecomposable (as a right module over itself) if and only if R has no nontrivial idempotents. A nonzero module M is said to be uniform if 16 every nonzero submodule of M is essential in M, equivalently, if every two nonzero submodules of M intersect nontrivially. M is said to be a uniserial module if all submodules of M are linearly ordered by inclusion.

A ring R is said to be a right (left ) valuation ring if RR (R R) is a uniserial module.