Topics in Abstract Algebra

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/ 00% May, 2010 Jacobson radical Chapter 2. /Commutative algebra/ 25% May, 2010 The spectrum of a ring, radical of an ideal Chapter 3. /Field theory/ 00% May, 2010 Galois theory, skew-field Chapter 4. /Linear algebra/ 00% May, 2010 The Moore-Penrose inverse, matrices over skew-fields, symplectic geometry, quadratic forms, smith normal form Chapter 5. /Lie algebras/ 00% May, 2010 Chapter 6. /Associative algebras/ 00% May, 2010 Radicals of non-commutative rings, separability, central simple algebras, Hopf algebras Part II. Applications /Banach algebras/ 00% May, 2010 Part III. Appendix /Sheaf theory/ 00% May, 2010 Algebra University level mathematics books A 50%