This is the second edition of a successful graduate textbook on modern algebra. The author has made several key additions to the content, whilst keeping all the material of the earlier edition. To the chapter on group theory, he has added new sections on abelian groups, finite abelian groups, solvable groups, nil potent groups and perfect groups. These sections are carefully inserted in the chapter to enhance with cohesion the explanation of the subject. Likewise in ring theory, there are new sections on special class of rings, quotient field, maximal and prime ideal. And in the chapter on vector spaces, inner product spaces and R-modules are now included. In each chapter the author has put in solved examples which both help to explain the topic under discussion and also test the understanding of that topic by the reader. Graded problems are also included to further assess the grasp of the subject. This book is a valuable text in modern algebra for graduate students, academics, and practising mathematicians. Its contents: Set Theory; Group Theory; Homomorphisms; Structure Theory of Group; Jordan-Holder Theorem and Solvable Groups; Ring Theory; Polynomial Rings; Factorization in Integral Domains; Vector Spaces; Linear Transformation; Field Theory; and, Key Points. This is the second edition of a successful text. Key additional content: solved examples and problems enhance understanding.