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Contributions to the Taxonomy of 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 Henry Chimal-Dzul April 2021 © 2021 Henry Chimal-Dzul. All Rights Reserved. 2 This dissertation titled Contributions to the Taxonomy of Rings by HENRY CHIMAL-DZUL has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio R. L´opez-Permouth Professor of Mathematics Florenz Plassmann Dean, College of Arts and Sciences 3 Abstract CHIMAL-DZUL, HENRY, Ph.D., April 2021, Mathematics Contributions to the Taxonomy of Rings (139 pp.) Director of Dissertation: Sergio R. L´opez-Permouth The present work aims to contribute to the taxonomy of various classes of non-commutative rings. These include the classes of reversible, reflexive, semicommutative, 2-primal, NI, abelian, Dedekind finite, Armendariz, and McCoy rings. In addition, two new families of rings are introduced, namely right real McCoy and polynomial semicommutative. The first contribution concerns the hierarchy and interconnections of reflexive, abelian, and semicommutative rings in the setting of finite rings. It is shown that a minimal reflexive abelian non-semicommutative ring has order 256 and that the group algebra F2D8 is an example of such a ring. This answers an open question in a paper on a taxonomy of finite 2-primal rings. The second set of contributions includes characterizations of reflexive, 2-primal, weakly 2-primal and NI rings in the setting of Morita context rings. The results on Morita context rings which are reflexive are shown to generalize known characterizations of prime and semiprime Morita context rings. Similarly, the results on 2-primal, weakly 2-primal and NI rings Morita context rings presented here generalize various known results for upper triangular matrix rings. Specially, a characterization of NI Morita context rings is shown to be equivalent to the (most) famous conjecture in Ring Theory: K¨othe's conjecture. The third collection of contributions addresses a study of right real McCoy rings and polynomial semicommutative rings; both classes of rings being introduced in this dissertation. The correlations between these types of rings is described, and a series of examples of finite polynomial semicommutative rings is given. Also, 4 equivalent conditions to the McCoy condition and some of its variations are also addressed. Finally, the last contribution establishes that the rings of column finite matrices, and row and column finite matrices over a reflexive ring are reflexive non-Dedekind finite. This result, along with some examples of finite non-reflexive Dedekind-finite rings that appear early in this dissertation, distinguishes the classes of reflexive and Dedekind-finite rings. 5 Dedication To My Parents who worked tirelessly to foster every aspect of my education in my youth 6 Acknowledgments I express my immense gratitude to my wife for constantly pushing me to challenge myself. Specially, I would like to thank her for her patience and support during this five years, and because she left a lot behind to undertake this journey with me in an unknown country. I would not be where I am now without her support. I express my heartfelt gratitude to my advisors, Professors Sergio L´opez-Permouth and Steve Szabo, for accepting me as his student and their constant support during these past five years. Because of their great sense of humor and insightful discussions in Mathematics, Sergio and Steve are persons who you instantly love and never forget once you meet them. I thank Professor Sergio for his exceptional Algebra courses full of \algebraic possums", for his amazing Coding Theory courses that made me love that field, for his pictorial way to teach me Quasi-Frobenius rings, for his multiple \colochito" and \kishpinudo" examples, and for his translations from \Yucateco" to \Chap´ın”.More importantly, for introducing me to Professor Steve through the Center of Ring Theory and its Applications. I thank Professor Steve for introducing me to some central topics in this dissertation, for always being available to chat in Skype hundreds of times a week, for asking more questions than I could answer, and for his great and unique vision decomposing rings using idempotents. Finally, in Steve and Sergio, I have found not only two great advisors, but two best friends on which I can deeply trust and always consider as members of my family. I am very grateful for all what I have learned from you. I thank my great friends and parents in USA, Dr. Erik Boczko and Nickie Bailes or as I call them with great affection \The Malitos". I thank Dr. Boczko for welcoming as his son in his house in January 2017 even I was a stranger to him. I 7 also thank him for his great discussions in Mathematics; for all his advise to succeed in my personal and professional life, and for sharing with me and my wife many adventures and UFC fights. I thank Nickie for always being there when I need help academically and personally. Their support made my life easier at Ohio University. To my Professors at Ohio University I owe my thanks for their wonderful lectures that strengthened my knowledge in Mathematics. Specially, to Professor Wei Lin for a great course in Statistics; Professor Archil Gullisashvili for his amazing lectures that made me to appreciate Real Analysis; Professor Vladimir Uspenskiy whose topolgy classes I was honored to attend; Professor Winfred Just for his outstanding lectures on Dynamical Systems and his interesting mathematical discussions inside and outside the classroom. Last but not least, to Ashley Pallone, Isaac Owusu-Mensah, Prabha Sherstha Shrestha, K. C. Rabi, and Rebi Muhammad (the \Big Big Big Boss"), for the privilege of sharing the 423 office at Morton Hall and all those big smiles that we had together. Also, I also must mention Aldo Garc´ıa-Guinto who became my little brother since the first day I met him at Ohio University. I will always remember each one of with great affection. 8 Table of Contents Page Abstract.......................................3 Dedication......................................5 Acknowledgments..................................6 Introduction..................................... 10 1 On Rings and Zero-divisors.......................... 16 1.1 Various Classes of Rings......................... 16 1.1.1 Symmetric Rings......................... 19 1.1.2 Reversible Rings......................... 20 1.1.3 Semicommutative Rings..................... 22 1.1.4 Reflexive Rings.......................... 24 1.1.5 Duo Rings............................. 26 1.2 Radicals and More Classes of Rings................... 28 1.2.1 PS -I and 2-primal Rings..................... 32 1.2.2 Weakly 2-primal, NI and NR Rings............... 34 1.3 Diagrams of Implications......................... 36 2 Minimal Reflexive non-Semicommutative Rings............... 40 2.1 Introduction................................ 40 2.2 Finite Local, Reflexive and Semicommutative Rings.......... 41 2.3 Small Abelian Reflexive Rings...................... 45 2.4 Examples of Abelian Reflexive non-Semicommutative Rings..... 51 3 Morita Context Rings............................. 61 3.1 Background on Morita Context Rings and Goals............ 61 3.2 Reflexive Morita Context Rings..................... 63 3.2.1 Reflexive Morita Context Rings................. 64 3.2.2 Semiprime Morita Context Rings................ 70 3.2.3 Prime Morita Context Rings................... 73 3.3 NI Morita Context Rings......................... 76 3.3.1 NI Morita Context Rings..................... 77 3.3.2 Weakly 2-primal and 2-primal Cases.............. 81 9 4 McCoy Rings and Polynomial Semicommutative Rings........... 84 4.1 An Overview on McCoy Rings...................... 84 4.2 Right Real McCoy Rings......................... 87 4.3 Abelian McCoy Rings........................... 92 4.4 Polynomial Semicommutative Rings................... 95 4.5 McCoy Conditions and Polynomial Rings................ 102 5 Reflexive non-Dedekind-finite Rings...................... 108 5.1 Row and Column Finite Matrices.................... 109 5.2 The Universal Property of Monoid Rings................ 111 5.3 A Monoid Ring Embedded in RCF M(R)................ 113 5.4 Reflexive Rings and Column Finite Matrix Rings........... 124 References...................................... 129 Alphabetical Index................................. 136 10 Introduction The concepts of \zero-divisor" and \nilpotent element" play an important role in Ring Theory. Various types of rings have been defined to distinguish between rings satisfying certain properties related to zero divisors and nilpotent elements, which in some senses are generalizations of properties satisfied by commutative rings. These type of rings include reduced, symmetric, reversible, reflexive, semicommutative, 2-primal, weakly 2-primal, NI, Armendariz, and McCoy rings. In addition to these, we also have prime, semiprime rings (which are examples of reflexive rings), and right duo rings (which are instances of semicommutative rings). The hierarchy and interconnections between these various types of rings have been topics of intensive research (e.g. [1], [2], [3], [4], [5], [6]). On one hand, a goal in this dissertation is to contribute to the hierarchy and interconnections between the various types of rings mentioned above in the setting of finite rings. On the other hand, it aims to characterize various of the ring classes aforementioned in the setting of Morita context
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