This book presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It uses semigroups and monoids as stepping stones to present the concepts of groups and rings. The author discusses the fundamentals of abstract algebra, before offering deeper coverage of group and ring theory. He also provides examples of abstract algebra concepts in matrices and calculus. The text contains numerous pedagogical elements, including exercises of varying levels of difficulty, chapter notes that point out variations in notation and approach, and study projects that cover an array of applications. A solutions manual is available for qualifying instructors.