Integral Closure

  • Irena SwansonEmail author


Since 2006, when the book on integral closures with Huneke and Swanson (Integral Closure of Ideals, Rings, and Modules. Cambridge University Press, Cambridge, 2006) was published, there has been more development in the area. This chapter is an attempt at catching up with that development as well as to fill in a few omissions. Some topics are worked out in detail whereas others are only outlined or mentioned.


Integral closure Rees valuations Computing integral closure Lipman–Sathaye theorem Multiplicity j-multiplicity Epsilon multiplicity Monomial ideals Goto numbers 

Subject Classifications:

13B22 Secondary: 13P05, 13P25, 13H15 


  1. 1.
    S. Barhoumi, H. Lombardi, An algorithm for the Traverso-Swan theorem on seminormal rings. J. Algebra 320, 1531–1542 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    C. Biviá-Ausina, Nondegenerate ideals in formal power series rings. Rocky Mt. J. Math. 34, 495–511 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    L. Bryant, Goto numbers of a numerical semigroup ring and the Gorensteiness of associated graded rings. Comm. Algebra 38, 2092–2128 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Corso, C. Polini, Links of prime ideals and their Rees algebras. J. Algebra 178, 224–238 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. Corso, C. Polini, W. Vasconcelos, Links of prime ideals. Math. Proc. Camb. Phil. Soc. 115, 431–436 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Corso, C. Huneke, W. Vasconcelos, On the integral closure of ideals. Manuscripta Math. 95, 331–347 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    S.D. Cutkosky, Asymptotic growth of saturated powers and epsilon multiplicity. Math. Res. Lett. 18, 93–106 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    T. de Jong, An algorithm for computing the integral closure. J. Symbolic Comput. 26, 273–277 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    S. Goto, Integral closedness of complete intersection ideals. J. Algebra 108, 151–160 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S. Goto, N. Matsuoka, R. Takahashi, Quasi-socle ideals in a Gorenstein local ring. J. Pure Appl. Algebra 212, 969–980 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    S. Goto, S. Kimura, N. Matsuoka, Quasi-socle ideals in Gorenstein numerical semigroup rings. J. Algebra 320, 276–293 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    S. Goto, S. Kimura, T.T. Phuong, H.L Truong, Quasi-socle ideals in Goto numbers of parameters. J. Pure Appl. Algebra 214, 501–511 (2010)Google Scholar
  13. 13.
    G.-M. Greuel, S. Laplagne, F. Seelisch, Normalization of rings. J. Symbolic Comput. 45(9), 887–901 (2010)Google Scholar
  14. 14.
    D. Grinberg, A few facts on integrality DETAILED VERSION.
  15. 15.
    W. Heinzer, I. Swanson, Goto numbers of parameter ideals. J. Algebra 321, 152–166 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95, 736–788 (1926)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    M. Hochster, Presentation depth and the Lipman-Sathaye Jacobian theorem, in The Roos Festschrift, vol. 2, Homology Homotopy Appl. 4, 295–314 (2002)Google Scholar
  18. 18.
    J. Horiuchi, Stability of quasi-socle ideals. J. Commut. Algebra 4, 269–279 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336 (Cambridge University Press, Cambridge, 2006)Google Scholar
  20. 20.
    J. Jeffries, J. Montaño, The j-multiplicity of monomial ideals. arXiv:math.AC/ 1212.1419 (preprint)Google Scholar
  21. 21.
    D. Katz, J. Validashti, Multiplicities and Rees valuations. Collect. Math. 61, 1–24 (2010) 2009.MathSciNetGoogle Scholar
  22. 22.
    D.A. Leonard, R. Pellikaan, Integral closures and weight functions over finite fields. Finite Fields Appl. 9, 479–504 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    H. Lombardi, Hidden constructions in abstract algebra. I. Integral dependence. J. Pure Appl. Algebra 167, 259–267 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    L.J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, (I). Am. J. Math. 91, 508–528 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    M. J. Saia, The integral closure of ideals and the Newton filtration.J. Algebraic Geom. 5, 1–11 (1996)Google Scholar
  26. 26.
    A. Seidenberg, Constructions in algebra. Trans. Am. Math. Soc. 197, 273–313 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    A.K. Singh, I. Swanson, Associated primes of local cohomology modules and of Frobenius powers. Int. Math. Res. Notices 30, 1703–1733 (2004)CrossRefMathSciNetGoogle Scholar
  28. 28.
    I. Swanson, Rees valuations, in Commutative Algebra: Noetherian and Non-Noetherian Perspectives, ed. by M. Fontana, S. Kabbaj, B. Olberding, I. Swanson (Springer, New York, 2010), pp. 421–440Google Scholar
  29. 29.
    B. Ulrich, J. Validashti, A criterion for integral dependence of modules. Math. Res. Lett. 15, 149–162 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    B. Ulrich, J. Validashti, Numerical criteria for integral dependence. Math. Proc. Camb. Phil. Soc. 151, 95–102 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    K.-i. Watanabe, K.-i. Yoshida, A variant of Wang’s theorem. J. Algebra 369, 129–145 (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Reed CollegePortlandUSA

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