Abstract
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.
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References
S. Barhoumi, H. Lombardi, An algorithm for the Traverso-Swan theorem on seminormal rings. J. Algebra 320, 1531–1542 (2008)
C. Biviá-Ausina, Nondegenerate ideals in formal power series rings. Rocky Mt. J. Math. 34, 495–511 (2004)
L. Bryant, Goto numbers of a numerical semigroup ring and the Gorensteiness of associated graded rings. Comm. Algebra 38, 2092–2128 (2010)
A. Corso, C. Polini, Links of prime ideals and their Rees algebras. J. Algebra 178, 224–238 (1995)
A. Corso, C. Polini, W. Vasconcelos, Links of prime ideals. Math. Proc. Camb. Phil. Soc. 115, 431–436 (1995)
A. Corso, C. Huneke, W. Vasconcelos, On the integral closure of ideals. Manuscripta Math. 95, 331–347 (1998)
S.D. Cutkosky, Asymptotic growth of saturated powers and epsilon multiplicity. Math. Res. Lett. 18, 93–106 (2011)
T. de Jong, An algorithm for computing the integral closure. J. Symbolic Comput. 26, 273–277 (1998)
S. Goto, Integral closedness of complete intersection ideals. J. Algebra 108, 151–160 (1987)
S. Goto, N. Matsuoka, R. Takahashi, Quasi-socle ideals in a Gorenstein local ring. J. Pure Appl. Algebra 212, 969–980 (2008)
S. Goto, S. Kimura, N. Matsuoka, Quasi-socle ideals in Gorenstein numerical semigroup rings. J. Algebra 320, 276–293 (2008)
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)
G.-M. Greuel, S. Laplagne, F. Seelisch, Normalization of rings. J. Symbolic Comput. 45(9), 887–901 (2010)
D. Grinberg, A few facts on integrality DETAILED VERSION. http://www.cip.ifi.lmu.de/~grinberg/Integrality.pdf
W. Heinzer, I. Swanson, Goto numbers of parameter ideals. J. Algebra 321, 152–166 (2009)
G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95, 736–788 (1926)
M. Hochster, Presentation depth and the Lipman-Sathaye Jacobian theorem, in The Roos Festschrift, vol. 2, Homology Homotopy Appl. 4, 295–314 (2002)
J. Horiuchi, Stability of quasi-socle ideals. J. Commut. Algebra 4, 269–279 (2012)
C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336 (Cambridge University Press, Cambridge, 2006)
J. Jeffries, J. Montaño, The j-multiplicity of monomial ideals. arXiv:math.AC/ 1212.1419 (preprint)
D. Katz, J. Validashti, Multiplicities and Rees valuations. Collect. Math. 61, 1–24 (2010) 2009.
D.A. Leonard, R. Pellikaan, Integral closures and weight functions over finite fields. Finite Fields Appl. 9, 479–504 (2003)
H. Lombardi, Hidden constructions in abstract algebra. I. Integral dependence. J. Pure Appl. Algebra 167, 259–267 (2002)
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)
M. J. Saia, The integral closure of ideals and the Newton filtration.J. Algebraic Geom. 5, 1–11 (1996)
A. Seidenberg, Constructions in algebra. Trans. Am. Math. Soc. 197, 273–313 (1974)
A.K. Singh, I. Swanson, Associated primes of local cohomology modules and of Frobenius powers. Int. Math. Res. Notices 30, 1703–1733 (2004)
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. 421–440
B. Ulrich, J. Validashti, A criterion for integral dependence of modules. Math. Res. Lett. 15, 149–162 (2008)
B. Ulrich, J. Validashti, Numerical criteria for integral dependence. Math. Proc. Camb. Phil. Soc. 151, 95–102 (2011)
K.-i. Watanabe, K.-i. Yoshida, A variant of Wang’s theorem. J. Algebra 369, 129–145 (2012)
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Swanson, I. (2014). Integral Closure. In: Fontana, M., Frisch, S., Glaz, S. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0925-4_19
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