Differential Identities in Rings and Algebras and Their Applications

The theory of differential identities in associative rings and algebras is the basis of this monograph. Informally, an identical relation involving arbitrary elements in the underlying rings (or algebras) along with the unknown differential function is called a differential identity in a ring (or algebra). Invariant theory, non-commutative geometry, mathematical physics, and the theory of rings and algebras are just a few of the fields where this abstract theory has proved to be an effective instrument for solving a wide range of challenging issues, and as the twenty-first century has arrived, the theory of differential identities has found enormous applications in resolving a number of unresolved problems in the theory of rings. This volume summarizes the findings and approaches that have significantly advanced the field during the previous three decades. The first chapter provides a brief introduction to the topic. The following three chapters cover the various kinds of derivations in rings and algebras as well as the interactions between the structure of some classes of rings with involution and the behavior of the underlying derivations, generalized derivations, skew derivations, and b-generalized derivations, as well as their corresponding properties. Chapter 5 explores the characterization of several kinds of higher derivable mappings and the structure of Lie and Jordan-type higher derivations. Although the book contains numerous applications of the conclusions presented in these chapters, the last chapter mostly focuses on the application of derivations. This research monograph is useful for researchers working in the area of differential identities in rings and algebras. It provides a comprehensive and authoritative account of research findings.

Shakir Ali is a Professor of Mathematics at the Aligarh Muslim University (AMU), Aligarh, India. He received his Ph.D. degree in Mathematics (2002) from AMU. He also worked as an Associate Professor at King Abdulaziz University, Jeddah, Saudi Arabia during December 2014 to August 2018. He is president of the Joint Indian-Indonesian-Iran-Malaysia-Turkey (IIIMT)-Algebra Forum. He was awarded TUBITAK (Turkey) and ISM-University Malaya (Malaysia) visiting professorships for the years 2019-2022 and 2023-2024, respectively. He is an author and co-author of more than 150 publications in reputed international journals and has been on research visits to several universities and institutions across the word to deliver invited\plenary\keynote lectures. Besides supervising several students for M. Phil/Ph.D. degrees in various disciplines of Mathematics, namely, Ring theory, Module Theory, Coding theory and Cryptography, Professor S. Ali has successfully completed many national and international research projects awarded by the University Grants Commission (UGC), Department of Science and Technology (DST), India, DSR-KAU, Kingdom of Saudi Arabia, TUBITAK, Goverment of Turkey. Presently, he is Editor / Managing Editor of several reputed international mathematical journals. He is also a co-editor of several research volumes and a life member of the American Mathematical Society, Indian Mathematical Society, Calcutta Mathematical Society, the Indian Science Congress Association, the Indian Society of Industrial and Applied Mathematics, the Ramanujan Mathematical Society.

Mohammad Ashraf is a retired Professor and a Former Chairperson, Department of Mathematics, Aligarh Muslim University, Aligarh, India. He is also a Former Dean, Faculty of Science, Aligarh Muslim University. He has published more than 250 peer-reviewed scientific articles in Internationally recognized and reputed journals. He has organized several international conferences and has edited Eight research Proceedings / Volumes. Besides supervising one and half dozen of students for Ph.D. degrees in various disciplines of Mathematics namely Ring theory, Linear Algebra, Coding theory and Graph theory, Prof. Ashraf has successfully completed many national and international research projects awarded by the University Grants Commission, Department of Science and Technology and National Board of Higher Mathematics. Currently, he is Editor / Managing Editor of many reputed international Mathematical journals. He is also a life member of the Indian Mathematical Society, Calcutta Mathematical Society, the Indian Science Congress Association, and the Indian Society of Industrial and Applied Mathematics.

Vincenzo De Filippis is a Professor of Mathematics at the University of Messina, Italy. He received his Ph.D. degree in Mathematics (1999) from the University of Messina, Italy. He is an author and co-author of more than 100 publications in professional journals and has been on research visits to several Institutions in Europe and Asia. His research interests include Ring Theory, Theory of Associative Algebras, and Linear and Multilinear Algebras. His theoretical work is primarily aimed at studying the structure of algebras satisfying functional identities. He has delivered many invited and plenary lectures at International Conferences on algebraic structures and their applications. He is also a co-editor of various research volumes and a member of the Italian Mathematical Society (UMI) and the National Society of Algebraic and Geometric Structures and their Applications (GNSAGA-INDAM).

Lahcen Oukhtite is a Professor at the Department of Mathematics, Faculty of Science and Technology, University Sidi Mohamed Ben Abdellah Fez (USMBA), Morocco. He is also Director of the laboratory of modelling and mathematical structures (USMBA). He received his PhD degree in mathematics (in 2000) from USMBA. He is an author and coauthor of more than 110 articles published in reputed international mathematical journals. He supervised more than 16 PhD theses and his research interests include rings theory, commutativity and structure of rings and near-rings, derivations in rings and Banach algebras, central simple algebras with involution, semi- simple algebras, graph Theory. He has organized several International conferences and edited four volumes and conference proceedings. He had visited several foreign countries to deliver plenary and invited lectures at international conferences and seminars. Professor Oukhtite is also member of editorial board of many international mathematical journals.

Nadeem ur Rehman is a Professor in the Department of Mathematics at Aligarh Muslim University (AMU), India. After earning his Ph.D. from AMU in 2000, he was awarded a DAAD postdoctoral fellowship in Germany. His teaching career began at the Birla Institute of Science and Technology, Pilani, before joining AMU in 2006. He was promoted to Associate Professor in 2015 and Professor in 2018, with a two-year tenure at Taibah University, KSA, from 2015 to 2017. Dr. Rehman has supervised numerous scholars, including ten Ph.D. students in areas such as Ring Theory, Coding Theory, and Graph Theory. He has presented research at international conferences and was a Visiting Scientist at the University of Maribor, Slovenia, under a bilateral exchange. With support from agencies like the DST, CSIR, and UGC, he has led major research projects and is an active reviewer for publishers like Elsevier, Springer, and AMS. A life member of several mathematical societies, he continues to contribute significantly to teaching and research.