This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

This textbook introduces the vast array of features and powerful mathematical functions of Mathematica using a multitude of clearly presented examples and worked-out problems. Each section starts with a description of a new topic and some basic examples. The author then demonstrates the use of new commands through three categories of problems - the first category highlights those essential parts of the text that demonstrate the use of new commands in Mathematica whilst solving each problem presented; - the second comprises problems that further demonstrate the use of commands previously introduced to tackle different situations; and - the third presents more challenging problems for further study. The intention is to enable the reader to learn from the codes, thus avoiding long and exhausting explanations. While based on a computer algebra course taught to undergraduate students of mathematics, science, engineering and finance, the book also includes chapters on calculus and solving equations, and graphics, thus covering all the basic topics in Mathematica. With its strong focus upon programming and problem solving, and an emphasis on using numerical problems that do not need any particular background in mathematics, this book is also ideal for self-study and as an introduction to researchers who wish to use Mathematica as a computational tool. This new edition has been extensively revised and updated, and includes new chapters with problems and worked examples.

The book offers a comprehensive introduction to Leavitt path algebras (LPAs) and graph C*-algebras. Highlighting their significant connection with classical K-theory-which plays an important role in mathematics and its related emerging fields-this book allows readers from diverse mathematical backgrounds to understand and appreciate these structures. The articles on LPAs are mostly of an expository nature and the ones dealing with K-theory provide new proofs and are accessible to interested students and beginners of the field. It is a useful resource for graduate students and researchers working in this field and related areas, such as C*-algebras and symbolic dynamics.