This present book comes from the first part of the lecture notes the author used for first-year graduate algebra course at the University of Minnesota, Purdue University and Peking University. The aim of this book is not only to give the student quick access to the basic knowledge of algebra, either for future advancement in the field of algebra, or for general background information, but also to show that algebra is truly a master key or "skeleton key" to many mathematical problems. The author wishes to present this book as an attempt to re-establish the contacts between algebra and other branches of mathematics and sciences. Many examples and exercises are included to illustrate the power of intuitive approaches to algebra.