Nadeem Ur Rehman - Böcker

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 applicationsof 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.

The volume explores the preservation properties of maps within complex algebraic structures. It comprises two parts. The first section focuses on the analysis of maps that preserve products within various algebraic frameworks, including semiprime rings, matrix algebras, and von Neumann algebras. It examines the conditions under which these maps maintain product structures, providing insights into their behaviour and implications in different algebraic settings. The discussions extend to the exploration of specific cases where these preservation properties are crucial, offering a detailed examination of how product-preserving maps influence the structural integrity of diverse algebraic entities. The second section shifts attention to the study of maps that preserve derivations, which play a critical role in the study of algebraic structures and their transformations. This part explores various forms of derivations and the maps that preserve them, highlighting their impact on the underlying algebraic structures. It delves into both linear and nonlinear contexts, examining how these maps affect the properties and behaviour of rings and algebras, offering valuable insights into the algebraic theory of derivations. The volume is a comprehensive resource for researchers and practitioners interested in the preservation properties of algebraic maps, contributing to advancing knowledge in the field of algebraic structures and their symmetries.Key Features:Explores advanced concepts in algebraic structuresCovers both product and derivation preservationComprehensive coverage of both additive and linear mapsProvides valuable theoretical and practical insightsEssential resource for advanced researchers.

The book intends to be a collection of research papers on algebra and related topics, most of which were presented at the international "Workshop on Associative Rings and Algebras with additional structures (WARA22)".