Integral closure has played a role in number theory and algebraic geometry since the nineteenth century but a modern formulation of the concept for ideals perhaps began with the work of Krull and Zariski in the 1930s. It has developed into a tool for the analysis of many algebraic and geometric problems. This book collects together the central notions of integral closure and presents a unified treatment. Techniques and topics covered include: behavior of the Noetherian property under integral closure analytically unramified rings the conductor field separability valuations Rees algebras Rees valuations reductions multiplicity mixed multiplicity joint reductions the BrianÃon-Skoda theorem Zariski's theory of integrally closed ideals in two-dimensional regular local rings computational aspects adjoints of ideals and normal homomorphisms. With many worked examples and exercises this book will provide graduate students and researchers in commutative algebra or ring theory with an approachable introduction leading into the current literature. Language: English