INJECTIVE MODULES OVER A PRTNCIPAL LEFT AND RIGHT IDEAL DOMAIN, \ryITH APPLICATIONS By Alina N. Duca SUBMITTED IN PARTTAL FULFILLMENT OF THE REQUIREMENTS FOR TI{E DEGREE OF DOCTOR OFPHILOSOPHY AT UMVERSITY OF MAMTOBA WINNIPEG, MANITOBA April 3,2007 @ Copyright by Alina N. Duca, 2007 THE T.TNIVERSITY OF MANITOBA FACULTY OF GRADUATE STI]DIES ***** COPYRIGHT PERMISSION INJECTIVE MODULES OVER A PRINCIPAL LEFT AND RIGHT IDEAL DOMAIN, WITH APPLICATIONS BY Alina N. Duca A Thesis/Practicum submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfïllment of the requirement of the degree DOCTOR OF PHILOSOPHY Alina N. Duca @ 2007 Permission has been granted to the Library of the University of Manitoba to lend or sell copies of this thesidpracticum, to the National Library of Canada to microfilm this thesis and to lend or sell copies of the film, and to University MicrofïIms Inc. to pubtish an abstract of this thesiVpracticum. This reproduction or copy of this thesis has been made available by authority of the copyright owner solely for the purpose of private study and research, and may only be reproduced and copied as permitted by copyright laws or with express written authorization from the copyright owner. UNIVERSITY OF MANITOBA DEPARTMENT OF MATIIEMATICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled "Injective Modules over a Principal Left and Right ldeal Domain, with Applications" by Alina N. Duca in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dated: April3.2007 External Examiner: K.R.Goodearl Research Supervisor: T.G.Kucera Examining Committee: G.Krause Examining Committee: W.Kocay T]NTVERSITY OF MANITOBA Date: April3,2007 Author: Alina N. Duca Title: Injective Modules over a Principal Left and Right ldeal Domain, with Applic ations Department: Mathematics Degree: PhD Convocation: May Year: 2007 Permission is herewith granted to University of Manitoba to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author TIIE AUTHOR RESERVES OTHER PTTBLICAITON RIGHTS, AND NEITIIER TTIE TIIESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTIIERWISE REPRODUCED \ryTTHOI.N THE A{.NHOR'S WRITTEN PERMISSION. TTM AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR TTIE USE OF ANY COPYRIGHTED MIffERIAL APPEARING IN THIS THESIS (OTITER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY IVRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED. ul IV I would like to dedicate this thesis to my loving husband Sebøstian. without his love, encourctgement, patience, and sacrifice, this thesis would not exist. I would also like to dedicate this thesis to my daughter Andrea, who I thankfor her understanding and well-timed distraction from møthematics. Abstract I study the (indecomposable) injective modules over some noetherian rings. The main results of this dissertation take place in the setting of a principal left and right ideat do- main .R. One indecomposable injective is the injective envelope (divisible hull) of the module n-8, and is isomorphic to the classical ring of quotients Q @r division algebra) of It. The other indecomposable injective modules are (up to isomorphism) in a one-to-one correspondence with the prime elements of the ring (up to similarity). Motivated by a classic treatment of O.Ore [31], I take advantage of the factorization theory in ,R and investigate the internal structure of an indecomposable injective E + Q. I describe it as a "layered" structure in two ways: first as the union of its socle series, and secondly, as the union of its elementary socle series, a concept from model theory. More powerful results concerning these objects are obtained via the technique of local- ization by considering their description over an extension fr of .R which is also a principal left and right ideal domain, but with a unique simple module (up to isomorphism). Each socle factor soc"(E) lsoc,r-1(,8) is a semisimple ffi-module, and I show that once a choice of basis at the level soc2 (-E)/soc1 (ø) is made, there is a canonical way to extend it through all levels so that the arithmetic of .Ð is understandable in terms of this basis. In addition,I analyze the right module structure of -E over its endomorphism ring and study the relationship with the elementary socle series of sE. Also, the bicommutator of As a consequence of all these results, I am able to classify and describe the indecom- posable injective modules over the first Weyl algebra (as well as over similar algebras), which-as wild modules-were thought to have an "unmanageable" structure [20]. Acknowledgments I would like to express my deepest gratitude to my advisor, Dr. Thomas Kucera for his invaluable support in every aspect of this work. In particular, I would like to thank him for his commitment to guiding me through my doctoral research, as well as for the time and endless patience while reading the various drafts of this thesis. His critical commentary on my work has played a major role in both the content and presentation of my discussion and arguments. I also owe special thanks to Dr. Günter Krause for his invaluable advice and inspiring discussions which have many times clarified my ideas. Sincere thanks also go to all the people of our department for their warm-hearted as- sistance during my studies. I would like to express my deep appreciation to my dear friends, Shirley and Timo Kangas, for their unconditional love, time and moral support. I am lucþ to have them as friends. Finally, I would like to give my warmest thanks to my parents, Alexandrina and Ioan Molie, and to my parents-in-law, Tereza and Cezar Duca, who continuously believed in me and encouraged me throughout the years. Contents Introduction Preliminaries RrNcs RNo MonuLES 1.1 Terminology and Notation. Basic Definitions 5 1.2 Injective Modules 7 1.3 Localization 13 1.4 Socle Series 22 1.5 E-adicCompletion. Bicommutator 24 MoIBI, THBony 1.6 Positive Primitive Formulas. pp-definable Subgroups 27 1.7 Elementary Socle Series 28 Principal Ideal Domains 33 2.1 Divisibility.Similarity 35 2.2 Factorization and Decomposition Theorems 48 Injective Modules over a Principal ldeal Domain 6T 3.1 Injective Modules over a Principal Ideal Domain 62 Table of Contents VIll 3.2 The Socle Series 64 3.3 The Elementary Socle Series 68 3.4 Localization 76 3.5 Bases for soc,,(q-E)/soc"-1(q E), n > 1 . 81 3.6 The Endomoqphism Ring and the Bicommutator of E . 88 Applications 93 4.1 The Indecomposable Injective Modules over ,k(ø)[g ,010r] 94 4.2 Injective Modules over the first V/eyl Algebra 97 4.3 Other Applications 103 Bibliography 106 lndex 110 Introduction Since the 1950s, there has been an increase in interest regarding non-commutative noethe- rian algebraic structures, mainly because of their appearance in various areas of mathemat- ics. The skew polynomial rings are one of the main classes of non-commutative noethe- rian rings, and in particular, rings of differential operators are one of the most important noncommutative noetherian algebras, with an important role in the representation theory of Lie algebras and in the algebraic analysis of systems of partial differential equations. The important role played by the injective modules over noetherian rings is obvious from the charactenzation of such rings via the injective modules, i.e., aring is left noethe- rian if and only if every injective left module is a direct sum of indecomposable injective modules, by 1271,1321. The problem of classifying and describing the indecomposable injective modules over a left Noetherian ring has been studied extensively in Jategaonkar [20], with a dominant role being played by the prime ideals of the ring, but also by the technique of localiza- tion. There, the indecomposable (uniform) injective modules have been classified as either 'tame' or 'wild'. The tame ones are uniquely determined by the prime ideals of the ring. Jategaonkar [20] haq made the remark that "wild modules are, as a rule, unmanageable", while "the tame ones are manageable". So far, there are few known descriptions of wild uniform injectives over noetherian (non-commutative) rings, so the study of such objects Introduction is indeed a very difÊcult task, if not almost impossible. This is especially unfortunate in the case of a non-commutative simple noetherian domain, where there are no prime ideals except the zero ideal. The only tame indecomposable injective module is the injective en- velope of the ring itself, the quotient field of the ring, all the other ones being wild. That is why the study of injective modules over the first Weyl algebra, the original motivation for this thesis, seemed to be such an unchallenged ground for research, yet with great potential interest. In the study of injective modules over a principal left and right ideal domain, we deal with two contradictory facts. The good news is that each wild indecomposable injective E is the union of its socle series-as a torsion module over an HNP ring ([20, A.1.4]). Tùe bad news, though, is that for each wild indecomposable injective module, the socle layers do not give additional information about the module itself past their semisimple structure. In some sense, a principal left and right ideal domain is 'almost commutative'. There is a relation of similarity between the elements of the ring allowing permutations of the order in a product by doing 'interchanges of factors'. Therefore, it is natural to expect that some of the structural properties of the indecomposable injectives over a commut¿tive noethçrian ring can be generalizedtoprincipal left andright ideal domains.