715

Advances in Rings and Modules

Sergio R. López-PermouthJae Keol ParkS. Tariq Rizvi

Cosmin S. RomanEditors

715

Advances in Rings and Modules

Sergio R. López-PermouthJae Keol ParkS. Tariq Rizvi

Cosmin S. RomanEditors

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Catherine Yan

2010 Mathematics Subject Classification. Primary 16Dxx, 16Sxx, 16Wxx, 16Exx, 16Gxx,16Nxx, 16Pxx, 16Uxx, 06Cxx, 15Bxx.

Library of Congress Cataloging-in-Publication Data

Names: Lopez-Permouth, S. R. (Sergio R.), 1957– editor.Title: Advances in rings and modules / Sergio R. Lopez-Permouth [and three others], editors.

Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Con-temporary mathematics ; volume 715 | Includes bibliographical references.

Identifiers: LCCN 2018007403 | ISBN 9781470435554 (alk. paper)Subjects: LCSH: Rings (Algebra) | Modules (Algebra) | AMS: Associative rings and algebras –

Modules, bimodules and ideals – Modules, bimodules and ideals. msc | Associative rings andalgebras – Rings and algebras arising under various constructions – Rings and algebras arisingunder various constructions. msc | Associative rings and algebras – Rings and algebras withadditional structure – Rings and algebras with additional structure. msc | Associative rings andalgebras – Homological methods – Homological methods. msc | Associative rings and algebras– Representation theory of rings and algebras – Representation theory of rings and algebras.msc | Associative rings and algebras – Radicals and radical properties of rings – Radicals andradical properties of rings. msc | Associative rings and algebras – Chain conditions, growthconditions, and other forms of finiteness – Chain conditions, growth conditions, and other formsof finiteness. msc | Associative rings and algebras – Conditions on elements – Conditions onelements. msc | Order, lattices, ordered algebraic structures – Modular lattices, complementedlattices – Modular lattices, complemented lattices. msc | Linear and multilinear algebra; matrixtheory – Special matrices – Special matrices. msc

Classification: LCC QA247 .A2824 2018 | DDC 512/.46–dc23 LC record available athttps://lccn.loc.gov/2018007403

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: https://doi.org/10.1090/conm/715

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10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18

This volume is dedicated to the memory of Professor Bruno J. Mullerfor his numerous contributions to the theory of Rings and Modules.

Bruno J. Muller (1934–2014)

Contents

Preface ix

Chains of semiprime and prime ideals in Leavitt path algebrasG.Abrams, B.Greenfeld, Z.Mesyan, and K.M.Rangaswamy 1

The conditions (Ci), i = 1, 2, 3, 11, 12, in rings, modules, categories,and lattices

Toma Albu 17

On K-semi discrete modulesAhmed Y. Ammar, Atif S. El-Araby, Mahmoud A. Kamal,

and Nawal S. Mahmoud 35

Nonlinear Lie triple higher derivation on triangular algebrasMohammad Ashraf and Aisha Jabeen 43

On universal localization of Noetherian ringsJohn A. Beachy and Christine M. Leroux 59

A survey of intrinsic extensions of ringsGary F. Birkenmeier and Enoch K. S. Lee 69

The reduced ring order and lower semi-latticesW.D.Burgess and R.Raphael 89

On a category of extensions whose endomorphism rings haveat most four maximal ideals

Federico Campanini and Alberto Facchini 107

Tilting cotorsion pairs and pure semisimple ringsNguyen Viet Dung and Jose Luis Garcıa 127

Twist invariants of graded algebrasK. R. Goodearl and M. T. Yakimov 141

Modules invariant under monomorphisms of their envelopesPedro A. Guil Asensio, Derya Keskin Tutuncu,

and Ashish K. Srivastava 171

Some results and questions on left-right symmetryDinh Van Huynh and Dinh Duc Tai 181

Rings in which every unit is a sum of a nilpotent and an idempotentArezou Karimi-Mansoub, Tamer Kosan, and Yiqiang Zhou 189

vii

viii CONTENTS

Commutators and anti-commutators of idempotents in ringsDinesh Khurana and T.Y. Lam 205

Distributive hierarchies of binary operationsSergio Lopez-Permouth and Louis H. Rowen 225

Idealizers in differential polynomial rings and generalized HNP ringsHidetoshi Marubayashi and Akira Ueda 243

Module hulls—similarities and contrastsJae Keol Park and S. Tariq Rizvi 255

Direct sums of completely almost self-injective modulesSurjeet Singh 273

Preface

Bruno J. Muller was a dear colleague and a mathematical inspiration to eachone of us. He was a leader in the field and a wise mentor and advisor to his doctoralstudents. Most of all, he was a dear friend, one that has been sorely missed sincehe passed away on October 27, 2014.

Based at McMaster University, Hamilton, Canada, and specializing in the the-ories of rings and modules, Bruno was a world-renowned mathematician. Amongother topics, Bruno did pioneering work on Localizations, Duality, and on the The-ory of Continuous and Discrete Modules.

He was an author of the first ever book (published by Cambridge UniversityPress) detailing the latest research developments in the theory of continuous, dis-crete and extending modules. Since its publication in 1990, that seminal bookbecame one of the most cited works in the area and continues today to be used byresearchers around the world. Professor Muller contributed fundamental researchpapers in many areas of algebra and impacted the discipline through his many PhDstudents at McMaster University.

Bruno was passionate about mathematics; he loved rings and modules andother algebraic structures. To recognize his contributions to Algebra, this volumecollects a series of contributions from members of the community presented herein remembrance of Bruno and as a tribute to his memory. We are thankful to allwho have participated for their enthusiastic embrace of the project and for theirvaluable contributions.

All papers in the volume are peer-refereed as usual. We express our gratitudeto all referees who helped in the review of the papers. The topics in the assortmentof the papers of this volume include Almost Self-Injective Modules, Commutatorsand Anti-Commutators of Idempotents, Derivations, Differential Polynomial Rings,Distributive Hierarchies of Binary Operations, Generalizations of Injectivity, HNP(Hereditary Noetherian Prime) Rings, Intrinsic Extensions of Rings, Leavitt PathAlgebras, Module Hulls, Module Invariants, Reduced Ring Orders, Tilting Cotor-sion Pairs, Twisted Invariants of Graded Algebras, and Universal Localizations.

As algebra continues to grow at a rapid pace, we hope the interesting resultspresented here will fuel and inspire young researchers for continued growth. We aresure Bruno would have enjoyed this volume and hope that you, the reader, will dothe same.

ix

x PREFACE

As always, we are indebted to the staff of the American Mathematical Societyfor their outstanding work. Special thanks are due, in particular, to Ms. ChristineThivierge for her tireless efforts to see this project to conclusion. We are thankful toproduction editor Mike Saitas for his dedication and professional efforts to producethe nice final form of the volume.

Sergio R. Lopez-PermouthJae Keol ParkS. Tariq Rizvi

Cosmin S. Roman

Selected Published Titles in This Series

715 Sergio R. Lopez-Permouth, Jae Keol Park, S. Tariq Rizvi, and CosminS. Roman, Editors, Advances in Rings and Modules, 2018

712 Nero Budur, Tommaso de Fernex, Roi Docampo, and Kevin Tucker, Editors,Local and Global Methods in Algebraic Geometry, 2018

711 Thomas Creutzig and Andrew R. Linshaw, Editors, Vertex Algebras andGeometry, 2018

710 Raphael Danchin, Reinhard Farwig, Jirı Neustupa, and Patrick Penel, Editors,Mathematical Analysis in Fluid Mechanics, 2018

709 Fernando Galaz-Garcıa, Juan Carlos Pardo Millan, and Pedro Solorzano,Editors, Contributions of Mexican Mathematicians Abroad in Pure and AppliedMathematics, 2018

708 Christian Ausoni, Kathryn Hess, Brenda Johnson, Ieke Moerdijk, and JeromeScherer, Editors, An Alpine Bouquet of Algebraic Topology, 2018

707 Nitya Kitchloo, Mona Merling, Jack Morava, Emily Riehl, and W. Stephen

Wilson, Editors, New Directions in Homotopy Theory, 2018

706 Yeonhyang Kim, Sivaram K. Narayan, Gabriel Picioroaga, and Eric S. Weber,Editors, Frames and Harmonic Analysis, 2018

705 Graham J. Leuschke, Frauke Bleher, Ralf Schiffler, and Dan Zacharia, Editors,Representations of Algebras, 2018

704 Alain Escassut, Cristina Perez-Garcia, and Khodr Shamseddine, Editors,Advances in Ultrametric Analysis, 2018

703 Andreas Malmendier and Tony Shaska, Editors, Higher Genus Curves inMathematical Physics and Arithmetic Geometry, 2018

702 Mark Grant, Gregory Lupton, and Lucile Vandembroucq, Editors, TopologicalComplexity and Related Topics, 2018

701 Joan-Carles Lario and V. Kumar Murty, Editors, Number Theory Related toModular Curves, 2018

700 Alexandre Girouard, Dmitry Jakobson, Michael Levitin, Nilima Nigam, IosifPolterovich, and Frederic Rochon, Editors, Geometric and Computational SpectralTheory, 2017

699 Mark L. Agranovsky, Matania Ben-Artzi, Catherine Beneteau, Lavi Karp,Dmitry Khavinson, Simeon Reich, David Shoikhet, Gilbert Weinstein,and Lawrence Zalcman, Editors, Complex Analysis and Dynamical Systems VII, 2017

698 Alexander M. Blokh, Leonid A. Bunimovich, Paul H. Jung, Lex G.Oversteegen, and Yakov G. Sinai, Editors, Dynamical Systems, Ergodic Theory, andProbability: in Memory of Kolya Chernov, 2017

697 Fabrizio Broglia, Francoise Delon, Max Dickmann, Danielle Gondard-Cozette,and Victoria Ann Powers, Editors, Ordered Algebraic Structures and Related Topics,2017

696 Ara S. Basmajian, Yair N. Minsky, and Alan W. Reid, Editors, In the Traditionof Ahlfors–Bers, VII, 2017

695 Katrina Barron, Elizabeth Jurisich, Antun Milas, and Kailash Misra, Editors,Lie Algebras, Vertex Operator Algebras, and Related Topics, 2017

694 Manjul Bhargava, Robert Guralnick, Gerhard Hiss, Klaus Lux, and Pham Huu

Tiep, Editors, Finite Simple Groups: Thirty Years of the Atlas and Beyond, 2017

693 Michael Cwikel and Mario Milman, Editors, Functional Analysis, HarmonicAnalysis, and Image Processing, 2017

692 Anatole Katok, Yakov Pesin, and Federico Rodriguez Hertz, Editors, ModernTheory of Dynamical Systems, 2017

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume, dedicated to Bruno J. Muller, a renowned algebraist, is a collection of papers

that provide a snapshot of the diversity of themes and applications that interest algebraists

today.

The papers highlight the latest progress in ring and module research and present work

done on the frontiers of the topics discussed.

In addition, selected expository articles are included to give algebraists and other math-

ematicians, including graduate students, an accessible introduction to areas that may be

outside their own expertise.

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