This book aims to cover algebraic structures and methods that play basic roles in other fields of mathematics such as algebraic geometry and representation theory. More precisely, the first chapter covers the rudiments of non-commutative rings and homological language that provide foundations for subsequent chapters. The second chapter covers commutative algebra, which we view as the local theory of algebraic geometry; the emphasis will be on (historical) connections to several complex variables. The third chapter is devoted to field theory, and the fourth to Linear algebra. The fifth chapter studies Lie algebra with emphasis on applications to arithmetic problems.

Contents

Part I. Foundations

Chapter 1. Non-commutative rings

Jacobson radical

Chapter 2. Commutative algebra

The spectrum of a ring, radical of an ideal

Chapter 3. Field theory

Galois theory, skew-field

Chapter 4. Linear algebra

Matrices over skew-fields, symplectic geometry, quadratic forms, smith normal form

Chapter 5. Lie algebras

Chapter 6. Associative algebras

Radicals of non-commutative rings, separability, central simple algebras, Hopf algebras

Part II. Applications

• Banach algebras

Part III. Appendix

• Sheaf theory