634 Noncommutative Rings and Their Applications

International Conference on Noncommutative Rings and Their Applications July 1–4, 2013 Université d’Artois, Lens, France

Steven Dougherty Alberto Facchini André Leroy Edmund Puczyłowski Patrick Solé Editors

American Mathematical Society Noncommutative Rings and Their Applications

International Conference on Noncommutative Rings and Their Applications July 1–4, 2013 Université d’Artois, Lens, France

Steven Dougherty Alberto Facchini André Leroy Edmund Puczyłowski Patrick Solé Editors

634

Noncommutative Rings and Their Applications

International Conference on Noncommutative Rings and Their Applications July 1–4, 2013 Université d’Artois, Lens, France

Steven Dougherty Alberto Facchini André Leroy Edmund Puczyłowski Patrick Solé Editors

American Mathematical Society Providence, Rhode Island Editorial Board of Contemporary Mathematics Dennis DeTurck, managing editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 16N60, 16P60, 16S34, 16S99, 16U99, 17B35, 94B05, 94B15, 94B60.

Library of Congress Cataloging-in-Publication Data International Conference on Noncommutative Rings and Their Applications (2013 : Artois, France) Noncommutative rings and their applications : International Conference on Noncommutative Rings and Their Applications, July 1–4, 2013, University of Artois, France / Steven Dougherty, Alberto Facchini, Andr´e Leroy, Edmund Puczylowski, Patrick Sol´e, editors. pages cm. – (Contemporary mathematics ; volume 634) Includes bibliographical references. ISBN 978-1-4704-1032-2 (alk. paper) 1. Noncommutative rings–Congresses. 2. Rings (Algebra)–Congresses. I. Dougherty, Steven, 1966– editor. II. Facchini, Alberto, editor. III. Leroy, Andr´e (Andr´e Gerard), 1955– editor. IV. Puczylowski, Edmund, editor. V. Sol´e, Patrick, 1960– editor. VI. Title. QA251.4.I58 2014 512.46–dc23 2014029564 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/634

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Preface vii List of Participants ix Leavitt Path Algebras of Cayley Graphs Arising from Cyclic Groups Gene Abrams and Benjamin Schoonmaker 1 Characterizations of Almost Injective Modules Adel Alahmadi, S. K. Jain, and Surjeet Singh 11 Automorphism-Invariant Modules Pedro A. Guil Asensio and Ashish K. Srivastava 19 Distributive Algebras, Isoclinism, and Invariant Probabilities Stephen M. Buckley 31

Ideals of the Enveloping Algebra U(sl3) Stefan Catoiu 53 On Second Submodules Sec¸il C¸eken and Mustafa Alkan 67 Open Problems in Coding Theory Steven T. Dougherty, Jon-Lark Kim, and Patrick Sole´ 79 Foundations of Algebraic Coding Theory Steven T. Dougherty 101

Counting Z2Z4-Additive Codes Steven T. Dougherty and Esengul¨ Salturk¨ 137 From Endomorphism Rings to Some Noteworthy Ideals in Categories of Modules Alberto Facchini 149 Lie and Jordan Properties in Group Algebras Edgar G. Goodaire and Cesar´ Polcino Milies 163 Rad-Projective δ Covers Yasser Ibrahim and Mohamed Yousif 175 On Lattices of Annihilators Malgorzata Jastrze¸bska and Jan Krempa 189

v vi CONTENTS

Clean Elements in Polynomial Rings Pramod Kanwar, AndreLeroy,´ and Jerzy Matczuk 197 On the Semiprime Smash Product Question Christian Lomp 205 Minimal Prime Ideals and Derivations Jerzy Matczuk 223 On Codes over Local Frobenius Non-Chain Rings of Order 16 Edgar Mart´ınez-Moro and Steve Szabo 227 On Quasi-Duo Rings E. R. Puczylowski 243 Isomorphism Invariants of Enveloping Algebras Hamid Usefi 253 Preface

The aim of the International Conference on Nonncommutative Rings and Their Applications, held in Lens, France, in July 2013 was to gather specialists in ring theory and in areas where this theory is used. The meeting gave the experts from different domains the opportunity to exchange their views, share their research, and learn from one another new results and problems in a friendly atmosphere. The works of the conference concerned, on one hand, classical aspects of ring and module theory and, on the other, some applications of this theory, focusing on coding theory, in which rings and modules play a more and more substantial role. The interplay between ring and coding theory was emphasized by Professor Steven Dougherty’s course in which he showed some of the ways (noncommutative) ring theory has influenced coding theory. The four invited speakers, Alberto Facchini, Dolors Herbera, Edmund Puczylowski, and Patrick Sol´e, presented their most recent works and many participants contributed to enrich further the themes addressed by this conference. This meeting was supported by the Laboratoire de Math´ematiques de Lens (LML), by different bodies from the Universit´e de Artois (RI, BQR), as well as by a regional organization (the F´ed´eration des Laboratoires de Math´ematiques du Nord pas de Calais). We would like to thank all the participants for their efforts and enthusiasm. Special thanks to Pramod Kanwar and Jerzy Matczuk who, during the conference, made practical matters go more smoothly. A very special thanks to the colleagues who kindly agreed to referee the pa- pers. Their expertise, promptitude, and professionalism improved the quality of the articles in this volume. Many thanks are due to the editorial staff of the AMS, in particular Christine Thivierge, for the very efficient way she managed the process of preparing and publishing these proceedings.

The Editors

vii

List of Participants

Gene Abrams Cristina Fernandez-Cordoba University of Colorado Universitat Aut`onoma de Barcelona Mustafa Alkan Walter Ferrer Akdeniz University Antalya Universidad de la Rep´ublica, Mamadou Barry Montevideo Universit´e Cheikh Anta Diop Dakar Yuval Ginosar Vladimir Bavula Haifa University Sheffield University Fatmanur Gursoy Mai Hoang Bien Yildiz Technical University, Istanbul Leiden University Dolors Herbera Geoffrey Booth Universitat Aut`onoma de Barcelona N. Mandela Metropolitan University Port Elizabeth Malgorzata Hryniewicka Bialystok University M’Hammed Boulagouaz Facult´e des Sciences et Techniques, F`es, Dijana Jakelic Morocco University of North Carolina, Wilmington and Stephen Buckley Max Planck Institute for Mathematics, National University of Ireland Bonn Maynooth David Jordan Marion Candau Sheffield University Universit´edeBretagneOccidentale Brest Pramod Kanwar Ohio University Stefan Catoiu De Paul University, Chicago Suat Karadeniz Ahmed Cherchem Fatih University, Istanbul Universit´eUSTHB,Alger Vladislav Kharchenko Nadeem Ahmed Dar National Autonomous University Aligarh Muslim University of Mexico (UNAM) Steven Dougherty Jan Krempa Scranton University Warsaw University Alberto Facchini Andr´e Leroy Padova University Universit´ed’Artois,Lens

ix x PARTICIPANTS

Christian Lomp Olfa Yemen Porto University Universit´e de Tunias El Manar Jerzy Matczuk Bahattin Yildiz Warsaw University Fatih University, Istanbul C´esar Polcino Milies Mohammed Yousif Universidade Federal do ABC and Ohio State University ˜ theUniversidadedeSAo Paulo Yiqiang Zhou Khanh Tung Nguyen Memorial University, Newfoundland University of Padova Michal Ziembowski Polytechnic University, Warsaw Zeynep Odemis-¨ Ozger¨ Fatih University, Istanbul Edmund Puczylowski Warsaw University Esengul Salturk Yildiz Technical University, Istanbul Ofir Schnabel Haifa University Javier Serda Universidade de S˜ao Paulo Daniel Smertnig University of Graz Patrick Sol´e Telecom Paris-Tech Djiby Sow Universit´eCheikAntaDiop Lukasz Swistek Warsaw University Steve Szabo Eastern Kentucky University Roger Ten Universitat Aut`onoma de Barcelona Felix Ulmer Universit´edeRennes Hamid Usefi Memorial University St John’s, Newfoundland Quimey Vivas Universidad de Buenos Aires Selected Published Titles in This Series

634 Steven Dougherty, Alberto Facchini, Andr´e Leroy, Edmund Puczylowski, and Patrick Sol´e, Editors, Noncommutative Rings and Their Applications, 2015 633 Delaram Kahrobaei and Vladimir Shpilrain, Editors, Algorithmic Problems of Group Theory, Their Complexity, and Applications to Cryptography, 2015 632 Gohar Kyureghyan, Gary L. Mullen, and Alexander Pott, Editors, Topics in Finite Fields, 2015 631 Siddhartha Bhattacharya, Tarun Das, Anish Ghosh, and Riddhi Shah, Editors, Recent Trends in Ergodic Theory and Dynamical Systems, 2015 630 Pierre Albin, Dmitry Jakobson, and Fr´ed´eric Rochon, Editors, Geometric and Spectral Analysis, 2014 629 Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, and Rub´ıE. Rodr´ıguez, Editors, Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, 2014 628 Anita T. Layton and Sarah D. Olson, Editors, Biological Fluid Dynamics: Modeling, Computations, and Applications, 2014 627 Krishnaswami Alladi, Frank Garvan, and Ae Ja Yee, Editors, Ramanujan 125, 2014 626 Veronika Furst, Keri A. Kornelson, and Eric S. Weber, Editors, Operator Methods in Wavelets, Tilings, and Frames, 2014 625 Alexander Barg and Oleg R. Musin, Editors, Discrete Geometry and Algebraic Combinatorics, 2014 624 Karl-Dieter Crisman and Michael A. Jones, Editors, The Mathematics of Decisions, Elections, and Games, 2014 623 Pramod N. Achar, Dijana Jakeli´c, Kailash C. Misra, and Milen Yakimov, Editors, Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, 2014 622 S. Ejaz Ahmed, Editor, Perspectives on Big Data Analysis, 2014 621 Ludmil Katzarkov, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, The Influence of Solomon Lefschetz in Geometry and Topology, 2014 620 Ulrike Tillmann, Søren Galatius, and Dev Sinha, Editors, Algebraic Topology: Applications and New Directions, 2014 619 Gershon Wolansky and Alexander J. Zaslavski, Editors, Variational and Optimal Control Problems on Unbounded Domains, 2014 618 Abba B. Gumel, Editor, Mathematics of Continuous and Discrete Dynamical Systems, 2014 617 Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang L¨uck, and J´erˆome Scherer, Editors, An Alpine Expedition through Algebraic Topology, 2014 616 G. L. Litvinov and S. N. Sergeev, Editors, Tropical and Idempotent Mathematics and Applications, 2014 615 Plamen Stefanov, Andr´as Vasy, and Maciej Zworski, Editors, Inverse Problems and Applications, 2014 614 James W. Cogdell, Freydoon Shahidi, and David Soudry, Editors, Automorphic Forms and Related Geometry, 2014 613 Stephan Stolz, Editor, Topology and Field Theories, 2014 612 Patricio Cifuentes, Jos´eGarc´ıa-Cuerva, Gustavo Garrig´os, Eugenio Hern´andez, Jos´eMar´ıa Martell, Javier Parcet, Keith M. Rogers, Alberto Ruiz, Fernando Soria, and Ana Vargas, Editors, Harmonic Analysis and Partial Differential Equations, 2014

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/. CONM 634 ocmuaieRnsadApplications and Rings Noncommutative

This volume contains the Proceedings of an International Conference on Noncommutative Rings and Their Applications, held July 1–4, 2013, at the Universite´ d’Artois, Lens, France. It presents recent developments in the theories of noncommutative rings and modules over such rings as well as applications of these to coding theory, enveloping algebras, and Leavitt path algebras. Material from the course “Foundations of Algebraic Coding Theory“, given by Steven Dougherty, is included and provides the reader with the history and background of cod- ing theory as well as the interplay between coding theory and algebra. In module theory, many new results related to (almost) injective modules, injective hulls and automorphism- invariant modules are presented. Broad generalizations of classical projective covers are studied and category theory is used to describe the structure of some modules. In some papers related to more classical ring theory such as quasi duo rings or clean elements, new points of view on classical conjectures and standard open problems are given. Descriptions of codes over local commutative Frobenius rings are discussed, and a list of open problems in coding theory is presented within their context. • oget ta. Editors al., et Dougherty

ISBN 978-1-4704-1032-2

9 781470 410322

CONM/634 AMS