Basic Modern Algebra with Applications

The book is primarily intended as a textbook on modern algebra for undergraduate mathematics students. It is also useful for those who are interested in supplementary reading at a higher level. The text is designed in such a way that it encourages independent thinking and motivates students towards further study. The book covers all major topics in group, ring, vector space and module theory that are usually contained in a standard modern algebra text.

In addition, it studies semigroup, group action, Hopf’s group, topological groups and Lie groups with their actions, applications of ring theory to algebraic geometry, and defines Zariski topology, as well as applications of module theory to structure theory of rings and homological algebra. Algebraic aspects of classical number theory and algebraic number theory are also discussed with an eye to developing modern cryptography. Topics on applications to algebraic topology, category theory, algebraic geometry, algebraic number theory, cryptography and theoretical computer science interlink the subject with different areas. Each chapter discusses individual topics, starting from the basics, with the help of illustrative examples. This comprehensive text with a broad variety of concepts, applications, examples, exercises and historical notes represents a valuable and unique resource.

Table of Contents

Chapter 1: Prerequisites: Basics of Set Theory and Integers

Chapter 2: Groups: Introductory Concepts

Chapter 3: Actions of Groups, Topological Groups and Semigroups

Chapter 4: Rings: Introductory Concepts

Chapter 5: Ideals of Rings: Introductory Concepts

Chapter 6: Factorization in Integral Domains and in Polynomial Rings

Chapter 7: Rings with Chain Conditions

Chapter 8: Vector Spaces

Chapter 9: Modules

Chapter 10: Algebraic Aspects of Number Theory

Chapter 11: Algebraic Numbers

Chapter 12: Introduction to Mathematical Cryptography

Appendix A: Some Aspects of Semirings

Appendix B: Category Theory

Appendix C: A Brief Historical Note

About the Authors

Mahima Ranjan Adhikari, PhD, is a former professor of pure mathematics at the University of Calcutta. His main interest lies is in algebra and topology. He has published a number of papers in several Indian and foreign journals including Proceedings of American Mathematical Society and four textbooks. 11 students have already been awarded PhD degree under his guidance. He is a member of American Mathematical Society and on the editorial board of several Indian and foreign journals. He was the president of the Mathematical Science Section of the 95th Indian Science Congress, 2008. He visited several institutions in India, USA, Japan, France, Greece, Sweden, Switzerland, Italy and many other countries on invitation. He is now the president of the Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC), www.imbic.org/index.html. He is also the principal investigator of an ongoing project, funded by the Government of India.

Avishek Adhikari, PhD, is an assistant professor of pure mathematics at the University of Calcutta. He is a recipient of the President of India Medal and Young Scientist Award. He was a post-doctorate fellow at the Research Institute INRIA, Rocquencourt, France. He was a visiting scientist at Indian Statistical Institute, Kolkata, and Linkoping University, Sweden. He visited many institutions in Japan, Sweden, France, England, Switzerland. His main interest lies is in algebra, discrete mathematics, theoretical computer science and their applications. He has published several papers in foreign journals and three textbooks on mathematics. He successfully completed several projects funded by the Government of India, and is a member of the research team from India for a collaborative Indo-Japan (DST-JST) research project. He is a member on editorial board of several journals and secretary (honorary) of the Research Institute IMBIC (www.imbic.org/index.html).