This book is the basis of teaching practice in the long term, the reference to a large number of domestic and foreign literature, textbooks, monographs, literature and research results to absorb one more to complete. This book chapters can be divided into three parts. The first part, both the basic concepts of the first chapter, which is based on the book, to be used in subsequent chapters, full attention should be. The second section includes the second and third chapters, describes the group with a theory of algebraic operations. The second chapter describes the group's most basic knowledge; the third chapter, further information on the normal subgroup and finite group, and the group associated with them the most basic of the most important theorems. The third part includes the fourth, fifth and sixth chapters, describes the ring with two algebraic operations with the domain theory. The fourth chapter describes the basics of ring; book courtesy of mathematicians, Chinese Academy of Sciences million Zhexian researchers and mathematicians in China, Chinese Academy of Sciences Professor Wang Zikun recommended publication, and courtesy of our mathematicians, Beijing Normal University PhD supervisor Professor Liu Shaoxue writing preface, the authors sincerely express their most sincere thanks! This book is the basis of teaching practice in the long term, the reference to a large number of domestic and foreign literature, textbooks, monographs, literature and research results to absorb one more to complete. This book chapters can be divided into three parts. The first part, both the basic concepts of the first chapter, which is based on the book, to be used in subsequent chapters, full attention should be. The second section includes the second and third chapters, describes the group with a theory of algebraic operations. The second chapter describes the group's most basic knowledge; the third chapter, further information on the normal subgroup and finite group, and the group associated with them the most basic of the most important theorems. The third part includes the fourth, fifth and sixth chapters, describes the ring with two algebraic operations with the domain theory. The fourth chapter describes the basics of ring; book courtesy of mathematicians, Chinese Academy of Sciences million Zhexian researchers and mathematicians in China, Chinese Academy of Sciences Professor Wang Zikun recommended publication, and courtesy of our mathematicians, Beijing Normal University PhD supervisor Professor Liu Shaoxue writing preface, the authors sincerely express their most sincere thanks!