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L-Stability in Rings and Left Quasi-Duo Rings

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L-Stability in Rings and Left Quasi-Duo Rings University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2018-05-28 L-stability in Rings and Left Quasi-duo Rings Horoub, Ayman Mohammad Abedalqader Horoub, A. M. A, (2018). L-stability in Rings and Left Quasi-duo Rings (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31958 http://hdl.handle.net/1880/106702 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY L-stability in Rings and Left Quasi-duo Rings by Ayman Mohammad Abedalqader Horoub A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS CALGARY, ALBERTA May, 2018 c Ayman Mohammad Abedalqader Horoub 2018 Abstract A ring R is said to have stable range 1 if, for any element a 2 R and any left ideal L of R, Ra + L = R implies a − u 2 L for some unit u in R. Here we insist only that this holds for all L in some non-empty set L(R) of left ideals of R, and say that R is left L-stable in this case. We say that a class C of rings is affordable if C is the class of left L-stable rings for some left idealtor L. In addition to the rings of stable range 1, it is known that the left uniquely generated rings and the rings with internal cancellation are both affordable. Here, we explore L-stability in general, derive some properties of this phenomenon and show that it captures many well-known results. This in turn yields new information about the left uniquely generated rings and the internally cancellable rings, and enables us to answer some open questions related to them. More importantly, we show that the directly finite rings are affordable, which gives a new perspective on these rings and the plethora of open questions related to them. Next, we turn to the class of left quasi-duo rings, that is, rings R in which every maximal left ideal is an ideal. But, surprisingly, these rings have many nice natural characterizations and properties that have passed unnoticed since they were introduced in 1995. Here we discuss this class of rings and prove some interesting new results for them. In particular, we introduce the notion of left width for rings, and use it to give a characterization of any left quasi-duo ring that has a finite left width. After that, a characterization of the I-finite left quasi-duo rings will be given. Finally, we introduce the left-max idealtor which is defined in terms of the maximal left ideals for any ring R. Then we study and characterize its associated rings which will be called the left- max stable rings. We also show that the class of left quasi-duo rings is neither left-max stable nor affordable. ii Acknowledgement I acknowledge with deep gratitude and appreciation my supervisor Professor W.K. Nicholson for his continuous support of my Ph.D study and related research and for his patience and moti- vation. His guidance helped me throughout my researching and writing of this thesis. I could not have imagined having a better supervisor for my Ph.D study. Especial thanks to Professor Berndt Brenken, Professor Michael Jacobson and Professor Mo- hamed Yousif from my Exam Committee for their valuable feedback and good comments. Grateful thanks to the University of Calgary for its remarkable facilities and for granting me the opportunity to pursue my Ph.D study. And many thanks to all people who helped me throughout my study in this University, especially, Professor Antony Ware and Yanmei Fei for their help. Finally, great thanks to my family; my parents, brothers and sisters for their continuous guidance and moral support throughout my academic life. Thanks for all your encouragement! iii To my parents, brothers and sisters for their encouragement and support. To my most favorite topic, \Rings". iv Table of Contents Page Abstract ii Acknowledgement iii Dedication iv Table of Contents v List of Symbols and Abbreviations vii I L-stability in Rings1 1 Introduction: Definitions, Examples and Fundamental Properties2 1.1 Importance of introducing the L-stability Notion . .2 1.2 Definitions and Four Major Examples . .3 1.3 Two Properties of Left Idealtors . 10 1.4 Basic Features and Essential Results for L-stability . 13 1.4.1 A Useful Tool to Check Ring Affordability . 14 1.4.2 DF Rings Revisited . 16 1.4.3 Special Left L-stable Elements . 17 1.4.4 A New Look at Three Old Well-Known Results . 19 1.4.5 An Extension of a Theorem of Bass . 22 2 Related Rings 24 2.1 Factor Rings . 24 2.2 Corner Rings . 27 2.3 Direct Products . 29 2.4 Subrings . 31 2.4.1 Motivation . 31 2.4.2 Ideal Extensions . 31 2.5 Polynomial Rings and Rings of Formal Power Series . 33 2.6 Matrix Rings . 35 2.6.1 L-stability is not a Morita Invariant Property of Rings in General . 36 2.6.2 Generalized Matrix Rings . 37 2.6.3 Generalized Upper Triangular Matrix Rings and Generalized Context-null Extensions . 39 v 3 Special Classes of Left Idealtors 43 3.1 Normal Left Idealtors . 43 3.2 Closed Left Idealtors . 44 3.3 Products of left L-stable elements . 46 3.4 Nice Left Idealtors . 50 II Left Quasi-duo Rings 54 4 Examples and Basic Results 55 4.1 Motivation . 55 4.2 Module Rings and a \Simple-Modules" Characterization of Left Quasi-duo Rings . 57 4.3 Examples and Elementary Properties . 59 4.4 A \Very Semisimple-Modules" Characterization of Left Quasi-duo Rings . 64 4.5 Generalized Upper Triangular Matrix Ring over Left Quasi-duo Rings . 66 5 Advanced Results for Left Quasi-duo Rings 68 5.1 Left Quasi-duo Rings with Finite Left Width . 68 5.1.1 The Width Theorem and the Left Width of Rings . 68 5.1.2 The Width Theorem for Left Quasi-duo Rings . 73 5.2 Left Quasi-duo Rings that are I-finite . 76 5.2.1 Left Quasi-duo I-finite Rings that are Semiprime . 76 5.2.2 Characterization of Left Quasi-duo I-finite Rings . 78 5.3 Left-Max idealtor and Left-Max stable Rings: . 81 Bibliography 85 Index 94 vi List of Symbols and Abbreviations ) implication , logical equivalence := or =: introducing definitions fx j P g the set of all x such that P (x) N set of natural numbers Z set of integers Q set of rational numbers R set of real numbers Zn integers modulo n a Z(p;q) Z(p;q) = f b 2 Q j p - b; q - bg for primes p; q in Z nZ = Zn subgroup (ideal) of Z generated by n [· disjoint union of sets =∼ isomorphic to RnA difference set A∗ Anf0g jAj cardinality of set A (number of elements in A) α : A ! B or A −!α B mapping α from A to B α(a) or aα used interchangeably for the image of a under mapping α a 7! aα definition of mapping α ,! inclusion mapping αβ or β ◦ α composite of mapping α and β (first α, then β ) 1A identity mapping on set A α−1 inverse of mapping α W (R); W (F ) Weyl algebra over R; over a field F R[x] ring of polynomials in x over R R[[x]] formal power series ring in x over R Mn(R) n × n matrices over a ring R Πi2I Ri direct product of the rings fRig hrii elements in Πi2I Ri RI direct product of I copies of ring R Rhxi j i 2 Ii free ring over a ring R generated by fxi j i 2 Ig πk canonical projections πk :Πi2I Ri ! Rk Mn[Ri;Vij] generalized n × n matrix ring over the rings Ri [ri; vij] elements of Mn[Ri;Vij] Tn[Ri;Vij] generalized n × n upper triangular matrix ring over the rings Ri vii T2[Ri;Vij] split-null extension (R; RVS, SWR, S) Morita context CNn[Ri;Vij] generalized n × n context-null extension hom(M; N) group of module homomorphisms end(M) endomorphism ring of left R-module M m¯ or m + N elements of the factor module M=N l(m) left annihilator of an element m in a left R-module M l(M) left annihilator of a left R-module M max N ⊆ RMN is a maximal submodule of a left R-module M ess N ⊆ RMN is an essential submodule of a left R-module M sm N ⊆ RMN is a small submodule of a left R-module M Rm principal left R-module generated by m S(M) or soc(M) socle of a left R-module M R=A factor ring r¯ or r + A elements of the factor ring R=A hai principal ideal generated by a A C RA is an ideal of a ring R A ⊆RRA is a left ideal of a ring R max A ⊆ RRA is a maximal left ideal of a ring R Sl(R) or S(RR) left socle of a ring R e idempotent element in a ring R, that is, e2 = e eRe corner ring with identity e2 = e a−1 multiplicative inverse of an element a in a ring R −a additive inverse of an element a in a ring R l(a) left annihilator of an element a in a ring R r(a) right annihilator of an element a in a ring R l(A) left annihilator of a left ideal A in a ring R r(A) right annihilator of a left ideal A in a ring R FF = F(R) = fA j A is left-max in Rg ∼ class(M) class(M) = fRN j RN = RMg fKi j i 2 Ig (SDR) system of distinct representatives of the isomorphism classes of ideal-simple left R-modules C(R) C = C(R) = fclass(Ki) j i 2 Ig, where fKi j i 2 Ig is an SDR wl(R) left width of a ring R J(R) Jacobson radical of a ring R D(R) set of idempotent elements in a ring R ureg(R) set of unit-regular elements in a ring R U(R) set of unit elements in a ring R Zr(R) right singular ideal of a ring R {∗} the class of ∗ rings; for example flocalg denotes the class of local rings L denotes an arbitrary left idealtor SR1 stable range 1 left UG left uniquely generated IC internal cancellation DF directly finite ¯ Kp Kaplansky's
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