JOURNAL BROWSE
Search
Advanced SearchSearch Tips
SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS
Hong, Joo-Youn; Lee, Hei-Sook; Noh, Sun-Sook;
  PDF(new window)
 Abstract
Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by . In this paper, if the m-adic order of P is 2, we show that . We also show that when w is the prime divisor associated to a simple v-ideal of order 2 and that w(R) = v(R) as well.
 Keywords
simple valuation ideal;order of an ideal;prime divisor;
 Language
English
 Cited by
1.
SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS,;

대한수학회논문집, 2008. vol.23. 4, pp.511-528 crossref(new window)
 References
1.
S. S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 70-99

2.
M. A. Hoskin, Zero-dimensional valuation ideals associated with plane curve branches, Proc. London Math. Soc. 6 (1956), no. 3, 70-99

3.
C. Huneke, Integrally closed ideals in two-dimensional regular local rings, Proc.Microprogram, in: Commutative Algebra, June 1987, MSRI Publication Series, vol. 15, Springer-Verlag, New York, 1989,325-337

4.
C. Huneke and J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500 crossref(new window)

5.
J. Lipman, Adjoints and polars of simple complete ideals in two-dimensional regular local rings, Bull. Soc. Math. Belgique 45 (1993), 223-244

6.
J. Lipman, Proximity inequalities for complete ideals in two-dimensional regular local rings, Contemp. Math. 159 (1994), 293-306 crossref(new window)

7.
S. Noh, The value semigroups of prime divisors of the second kind on 2-dimensional regular local rings, Trans. Amer. Math. Soc. 336 (1993), 607-619 crossref(new window)

8.
S. Noh, Sequence of valuation ideals of prime divisors of the second kind in 2-dimensional regular local rings, J. Algebra 158 (1993), 31-49 crossref(new window)

9.
S. Noh, Adjacent integrally closed ideals in dimension two, J. Pure Appl. Algebra 85 (1993), 163-184 crossref(new window)

10.
S. Noh, Valuation ideals of order one in two-dimensional regular local rings, Comm. Algebra 28 (2000), no. 2, 613-624 crossref(new window)

11.
S. Noh, Valuation ideals of order two in 2-dimensional regular local rings, Math. Nachr. 261-262 (2003), 123-140

12.
P. Ribenboim, The theory of classical valuations, Springer-Verlag, New York, 1999

13.
O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938), 151-204 crossref(new window)

14.
O. Zariski and P. Samuel, Commutative Algebra, vol. 2, D. Van Nostrand, Princeton, 1960