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Integral Closure of Ideals, Rings, and Modules

Integral Closure of Ideals, Rings, and Modules

$101.00 (R)

Part of London Mathematical Society Lecture Note Series

  • Date Published: October 2006
  • availability: Available
  • format: Paperback
  • isbn: 9780521688604

$ 101.00 (R)

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About the Authors
  • 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 Briançon-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.

    • First book to collect the material on integral closures into a unified treatment
    • Ideal for graduate students and researchers in commutative algebra or ring theory, with many worked examples and exercises
    • Provides a one-stop shop for newcomers and experts
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    Reviews & endorsements

    "...the appearance of this wonderful text, which gives an extensive, detailed and pretty well complete account of this whole area up to the present day, is very welcome. As is expected of these authors, the treatment is impressively well-informed, wide-ranging and convincing in its treatment of all aspects of the subject. The mathematics is presented elegantly and efficiently, helpful motivation is put in place in a perfectly judged manner, striking and informative asides are mentioned throughout, and the wealth of background and general culture carried by the many and varied exercises invaluable. ...A huge amount of material scattered throughout the classical and mroe up-to-date literature has been brought together in detailed, coherent and thoroughly worked-through form." - Liam O'Carroll, Mathematical Reviews Clippings

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    Product details

    • Date Published: October 2006
    • format: Paperback
    • isbn: 9780521688604
    • length: 448 pages
    • dimensions: 229 x 152 x 25 mm
    • weight: 0.611kg
    • contains: 6 b/w illus. 346 exercises
    • availability: Available
  • Table of Contents

    Table of basic properties
    Notation and basic definitions
    1. What is the integral closure
    2. Integral closure of rings
    3. Separability
    4. Noetherian rings
    5. Rees algebras
    6. Valuations
    7. Derivations
    8. Reductions
    9. Analytically unramified rings
    10. Rees valuations
    11. Multiplicity and integral closure
    12. The conductor
    13. The Briançon-Skoda theorem
    14. Two-dimensional regular local rings
    15. Computing the integral closure
    16. Integral dependence of modules
    17. Joint reductions
    18. Adjoints of ideals
    19. Normal homomorphisms
    Appendix A. Some background material
    Appendix B. Height and dimension formulas

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    Integral Closure of Ideals, Rings, and Modules

    Irena Swanson, Craig Huneke

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  • Authors

    Irena Swanson, Reed College, Portland
    Irena Swanson is a Professor in the Department of Mathematics at Reed College, Portland.

    Craig Huneke, University of Kansas
    Craig Huneke is the Henry J. Bischoff Professor in the Department of Mathematics, University of Kansas.

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