Springer Proceedings in Mathematics & Statistics

Volume 174 Springer Proceedings in Mathematics & Statistics

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533 Syed Tariq Rizvi • Asma Ali Vincenzo De Filippis Editors

Algebra and its Applications ICAA, Aligarh, India, December 2014

123 Editors Syed Tariq Rizvi Vincenzo De Filippis Department of Mathematics Department of Mathematics and Computer The Ohio State University Science Lima, OH University of Messina USA Messina Italy Asma Ali Department of Mathematics Aligarh Muslim University Aligarh, Uttar Pradesh India

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-10-1650-9 ISBN 978-981-10-1651-6 (eBook) DOI 10.1007/978-981-10-1651-6

Library of Congress Control Number: 2016943788

Mathematics Subject Classification (2010): 08-xx, 13-xx, 16-xx, 17-xx, 20-xx

© Springer Science+Business Media Singapore 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

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This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd. Preface

The international conference on “Algebra and its Applications” was organized by the Department of Mathematics, Aligarh Muslim University, Aligarh, India, and was held during December 15–17, 2014, under the UGC-DRS (SAP-II) pro- gramme. The conference was sponsored by Aligarh Muslim University (AMU), Science and Engineering Research Board (SERB), Department of Science and Technology (DST), New Delhi and the National Board of Higher Mathematics (NBHM), Mumbai. The purpose of the conference was to bring together algebraists from all over the world working in Algebra and related areas to present their recent research works, exchange new ideas, discuss challenging issues and foster future collaborations in Algebra and is applications. An important aim of the conference was to expose young researchers to new research developments and ideas in Algebra via the talks presented and the research interactions the conference provided. This research volume based on the proceedings of the conference consists of research literature on latest developments in various branches of algebra. It is the outcome of the invited lectures and research papers presented at the conference. It also includes some articles by invited algebraists who could not attend the con- ference. This includes Professors Jae Keol Park, Busan National University, Busan, South Korea; Akihiro Yamamura, Akita University, Japan; Shuliang Huang, Chuzhou University, China; Shervin Sahebi and V. Rahmani, Islamic Azad University, Tehran, Iran; Shreedevi K. Masuti and Parangama Sarkar, IIT Bombay, Mumbai; C. Selvaraj, Periyar University, Salem, Tamil Nadu; T. Tamizh Chelvam, Sundarnar University, Tamil Nadu; S. Tariq Rizvi, The Ohio State University, Ohio, Lima, USA; N.K. Thakare, Pune University, Pune; A. Tamilselvi, Ramanujan Institute for Advanced Study in Mathematics, Chennai; and V.S. Kapil, Himachal Pradesh University, Shimla. To maintain the quality of the work, all papers of the research volume are peer-reviewed by global subject experts. As Algebra continues to experience

v vi Preface tremendous growth and diversification, these articles highlight the cross-fertilization of ideas between various branches of algebra and exhibit the latest methods and techniques needed in solving a number of existing research problems while provide new open questions for further research investigations. These will cover a broad range of topics and variety of methodologies. It is expected that this research volume will be a valuable resource for young as well as experienced researchers in Algebra. Professor Asma Ali was the convener of the conference and Professor M. Mursaleen, the coordinator of the DRS programme. The conference had Professor Patrick W. Keef, Whitman College, Walla, Walla, USA, the chief guest and Professor Ashish K. Srivastava, Saint Louis University, St. Louis, USA, the guest of honor. The enriching programme contained a keynote address on computer-aided linear algebra by no less a mathematician than Professor Vasudevan Srinivas of a premier research centre of India, the Tata Institute of Fundamental Research (TIFR). Professor Vasudevan is also the recipient of Indian National Science Academy (INSA) Medal for Young Scientists, elected Fellow of Indian Academy of Science (IAS) and has received the B.M. Birla Science Award, Swarnajayanthi Fellowship Award, Bhatnagar Prize, J.C. Bose Fellowship, and TWAS Mathematics Prize. A total of 13 plenary talks and 20 invited talks on current topics of algebra and its applications were delivered by distinguished algebraists. The speakers included, Professors Luisa Carini; Vincenzo De Filippis, Italy; Nanqing Ding, China; Sudhir R. Ghorpade, IIT Bombay; Jugal K. Verma, Tony Joseph and Ananthanarayan, IIT Bombay; Manoj Kummini, Chennai Mathematical Society, Chennai; Sarang Sane, Indian Institute of Science, Bangalore; Kapil Hari Paranjape, IISER Mohali, Chandigarh; M.K. Sen, University of Calcutta; B.N. Waphare, University of Pune, Pune; A.R. Rajan, University of Kerala, Kerala; B.M. Pandeya, Banaras Hindu University; P.G. Romeo, Cochin University of Sciences and Technology (CUSAT), Kerala; Manoj Kumar Yadav, Harish-Chandra Research Institute (HRI), Allahabad; R.P. Sharma, Himachal Pradesh University, Summerhill, Shimla, and others. Another 44 research papers were presented by young researchers in algebra. Overall, the conference was greatly successful in its aims and objectives. The organizing committee, for the first time in mathematics conferences held at A.M.U. Aligarh, initiated the best paper presentation award. The award was initi- ated to motivate and inspire the young talents below the age of 32 years, and carried a participation certificate and a modest prize in cash. We thank all our colleagues who contributed papers to this research volume and those who graciously accepted to serve as referees of the submitted papers. We also thank the organizing team, the members of the department and the student workers who actively helped in making the conference a success. The conference could not be so successful without their active help and participation. The financial support Preface vii from all agencies listed is gratefully acknowledged. We also express our thanks to Springer for bringing out this volume in a nice form. The professional help and cooperation provided by Mr. Shamim Ahmad, Editor (Math. Sciences) is thankfully appreciated and acknowledged.

Lima, USA Syed Tariq Rizvi Aligarh, India Asma Ali Messina, Italy Vincenzo De Filippis Contents

On Some Classes of Module Hulls ...... 1 Jae Keol Park and S. Tariq Rizvi Spined Product Decompositions of Orthocryptogroups ...... 31 Akihiro Yamamura Generalized Skew Derivations and g-Lie Derivations of Prime Rings ...... 45 Vincenzo De Filippis Additive Representations of Elements in Rings: A Survey...... 59 Ashish K. Srivastava Notes on Commutativity of Prime Rings ...... 75 Shuliang Huang Generalized Derivations on Rings and Banach Algebras ...... 81 Shervin Sahebi and Venus Rahmani A Study of Suslin Matrices: Their Properties and Uses...... 89 Ravi A. Rao and Selby Jose Variations on the Grothendieck–Serre Formula for Hilbert Functions and Their Applications ...... 123 Shreedevi K. Masuti, Parangama Sarkar and J.K. Verma de Rham Cohomology of Local Cohomology Modules ...... 159 Tony J. Puthenpurakal Central Quotient Versus Commutator Subgroup of Groups ...... 183 Manoj K. Yadav

ix x Contents

Robinson–Schensted Correspondence for the Walled Brauer Algebras and the Walled Signed Brauer Algebras ...... 195 A. Tamilselvi, A. Vidhya and B. Kethesan C-Semigroups: A Survey ...... 225 M.K. Sen and S. Chattopadhyay Comparability Axioms in Orthomodular Lattices and Rings with Involution ...... 241 N.K. Thakare, B.N. Waphare and Avinash Patil Structure Theory of Regular Semigroups Using Categories...... 259 A.R. Rajan Biorder Ideals and Regular Rings ...... 265 P.G. Romeo and R. Akhila Products of Generalized Semiderivations of Prime Near Rings ...... 275 Asma Ali and Farhat Ali n-Strongly Gorenstein Projective and Injective Complexes ...... 293 C. Selvaraj and R. Saravanan Generalized Derivations with Nilpotent Values on Multilinear Polynomials in Prime Rings...... 307 Basudeb Dhara Properties of Semi-Projective Modules and their Endomorphism Rings...... 321 Manoj Kumar Patel ƒ! ! Labeling of Sets Under the Actions of Sn and An ...... 329 Ram Parkash Sharma, Rajni Parmar and V.S. Kapil Zero-Divisor Graphs of Laurent Polynomials and Laurent Power Series ...... 345 Anil Khairnar and B.N. Waphare Pair of Generalized Derivations and Lie Ideals in Prime Rings ...... 351 Basudeb Dhara, Asma Ali and Shahoor Khan On Domination in Graphs from Commutative Rings: A Survey ...... 363 T. Tamizh Chelvam, T. Asir and K. Selvakumar On Iso-Retractable Modules and Rings ...... 381 A.K. Chaturvedi Normal Categories from Completely Simple Semigroups ...... 387 P.A. Azeef Muhammed Contents xi

Ordered Semigroups Characterized in Terms of Intuitionistic Fuzzy Ideals ...... 397 Noor Mohammad Khan and Mohammad Aasim Khan On a Problem of Satyanarayana Regarding the Recognizability of Codes...... 421 R.D. Giri Editors and Contributors

About the Editors

Syed Tariq Rizvi is Professor and the Coordinator of Department of Mathematics, the Ohio State University (OSU), Lima, USA. After receiving his Ph.D. from McMaster University (Canada) in 1981 and teaching at the University of Waterloo (Canada), he joined OSU in 1982. A recipient of the Outstanding Scholar Award from OSU and a Fulbright Fellow Award from the Fulbright Board, Washington, DC, Professor Rizvi is world renowned for his research contributions to the theory of rings and modules. He has numerous research papers to his credit published in several international journals of high repute, published a monograph by Birkhäuser and coedited five research volumes with other reputed publishers. He also serves on editorial boards of ten international journals, including two as a managing or executive editor. Professor Rizvi has given over 120 invited talks at several national and international conferences, many as a plenary speaker. He has regularly orga- nized sessions at a number of conferences including at the American Mathematical Society conferences and the Ohio State-Denison Math Conference series. Asma Ali is Professor of Mathematics at the Department of Mathematics, Aligarh Muslim University (AMU), Aligarh, India. She has made valuable contributions to mathematical research with publishing a number of papers in international journals of repute. She has been awarded the honor of recognition for the achievements as a successful professional at the International Women’s Day at AMU in 2015. She is also life member of many organizations including the American Mathematical Society, the Indian Mathematical Society and the Indian Science Congress. Vincenzo De Filippis is Associate Professor of Mathematics at the University of Messina, Italy. He received the M.S. (1993) and Ph.D. (1999) degrees from the University of Messina. He is an author and coauthor of numerous professional

xiii xiv Editors and Contributors publications and has been on research visits to several institutions in Europe, India, China and Turkey. His research interests include ring theory, theory of associative algebras, linear and multilinear algebra. His theoretical work is primarily aimed at studying the structure of algebras satisfying functional identities.

Contributors

R. Akhila Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, India Farhat Ali Department of Mathematics, Aligarh Muslim University, Aligarh, India T. Asir Department of Mathematics (DDE), Madurai Kamaraj University, Madurai, Tamil Nadu, India P.A. Azeef Muhammed Department of Mathematics, University of Kerala, Trivandrum, Kerala, India S. Chattopadhyay Sovarani Memorial College, Jagathballavpur, India A.K. Chaturvedi Department of Mathematics, University of Allahabad, Allahabad, India Basudeb Dhara Department of Mathematics, Belda College, Belda Paschim Medinipur, West Bengal, India Basudeb Dhara Department of Mathematics, Belda College, Paschim Medinipur, India R.D. Giri RTM Nagpur University, Nagpur, India Shuliang Huang School of Mathematics and Finance, Chuzhou University, Chuzhou, People’s Republic of China Selby Jose School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India; Department of Mathematics, Institute of Science, Mumbai, India V.S. Kapil Department of Mathematics, Himachal Pradesh University, Shimla, India B. Kethesan Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India Anil Khairnar Department of Mathematics, Abasaheb Garware College, Pune, India Mohammad Aasim Khan Department of Mathematics, Aligarh Muslim University, Aligarh, India Editors and Contributors xv

Noor Mohammad Khan Department of Mathematics, Aligarh Muslim University, Aligarh, India Shahoor Khan Department of Mathematics, Aligarh Muslim University, Aligarh, India Shreedevi K. Masuti Chennai Mathematical Institute, Chennai, Tamil Nadu, India Jae Keol Park Department of Mathematics, Busan National University, Busan, South Korea Rajni Parmar Department of Mathematics, Himachal Pradesh University, Shimla, India Manoj Kumar Patel Department of Mathematics, NIT Nagaland, Dimapur, India Avinash Patil Garware College of Commerce, Pune, India Tony J. Puthenpurakal Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, India Venus Rahmani Department Of Mathematics, Islamic Azad University, Tehran, Iran A.R. Rajan Department of Mathematics, University of Kerala, Kariavattom, India Ravi A. Rao School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India; Department of Mathematics, Institute of Science, Mumbai, India P.G. Romeo Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, India Shervin Sahebi Department of Mathematics, Islamic Azad University, Tehran, Iran R. Saravanan Department of Mathematics, Periyar University, Salem, Tamil Nadu, India Parangama Sarkar Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India K. Selvakumar Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India C. Selvaraj Department of Mathematics, Periyar University, Salem, Tamil Nadu, India M.K. Sen Department of Pure Mathematics, University of Calcutta, Kolkata, India Ram Parkash Sharma Department of Mathematics, Himachal Pradesh University, Shimla, India xvi Editors and Contributors

Ashish K. Srivastava Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO, USA A. Tamilselvi Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India T. Tamizh Chelvam Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India N.K. Thakare Dhule, India J.K. Verma Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India A. Vidhya Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India B.N. Waphare Department of Mathematics, Center for Advanced Studies in Mathematics, Savitribai Phule Pune University, Pune, India Manoj K. Yadav School of Mathematics, Harish-Chandra Research Institute, Jhunsi, Allahabad, India Akihiro Yamamura Department of Mathematical Science and Electrical- Electronic-Computer Engineering, Akita University, Akita, Japan