The title of the book is self-explanatory. The first five chapters cover the basics; groups, rings, modules, vector spaces, and fields (including the Galois theory). The remaining three chapters are independent and can be treated as optional in a one-year course. They cover non-commutative rings, group extensions, and abelian groups. There is no introductory chapter reviewing what-everyone-reading-this-book-should-know. The corresponding material is placed where it is needed. There are many exercises throughout the text and there are problem sets at the end of each chapter. A novelty is an early introduction of the tensor product and the concept of projective modules. These concepts are used to simplify some standard proofs. The book can serve well as a basis for a one year graduate course.