This textbook offers an elementary introduction to noncommutative rings and algebras. Beginning with the classical theory of finite-dimensional algebras, it then develops a more general structure theory of rings, grounded in modules and tensor products. The final chapters cover free algebras, polynomial identities, and rings of quotients. Many results are presented in a simplified form rather than in full generality, with an emphasis on clear and understandable exposition. Prerequisites are kept to a minimum, and new concepts are introduced gradually and carefully motivated. Introduction to Noncommutative Algebra is thus accessible to a broad mathematical audience, though it is primarily intended for beginning graduate students and advanced undergraduates encountering the subject for the first time. This new edition includes several additions and improvements, while preserving the original text’s character and approach. Praise for the first edition: “It will soon find its place in classrooms” — Plamen Koshlukov, Mathematical Reviews “Very well written [...] very pleasant to read” — Veereshwar A. Hiremath, zbMATH “An excellent choice for a first graduate course” — D. S. Larson, Choice