Abstract Algebra

Abstract algebra, a broad division of mathematics, is the study of algebraic structures. Linear algebra, elementary number theory, and discrete mathematics are sometimes considered branches of abstract algebra. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebra over a field. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a powerful formalism for analyzing and comparing different algebraic structures. Universal algebra is a related subject that studies the nature and theories of various types of algebraic structures as a whole. For example, universal algebra studies the overall theory of groups, as distinguished from studying particular groups. This book, Abstract Algebra, is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. Important branches of abstract algebra are commutative algebra, representation theory, and homological algebra.