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SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS
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 Title & Authors
SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS
Noh, Sun-Sook;
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 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 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 -ideals and all the other -ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple -ideal P is either simple or the product of two simple -ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple -ideals when P is satellite of order 3 in terms of the invariant , where is the prime divisor associated to P and m = (x, y). Denote by b and let b = 3k + 1 for k = 0, 1, 2. Let be the number of nonmaximal simple -ideals of order i for i = 1, 2, 3. We show that the numbers = (, , ) = (, 1, 1) and that the rank of P is = k + 3. We then describe all the -ideals from m to P as products of those simple -ideals. In particular, we find the conductor ideal and the -predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for . We also find the value semigroup of a satellite simple valuation ideal P of order 3 in terms of .
 Keywords
simple valuation ideal;order of an ideal;prime divisor;proximity of simple integrally closed ideal;regular local ring;
 Language
English
 Cited by
 References
1.
S. S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 70-99 crossref(new window)

2.
J. Hong, H. Lee, and S. Noh, Simple valuation ideals of order two in 2-dimensional regular local rings, Commun. Korean Math. Soc. 20 (2005), no. 3, 427-436 crossref(new window)

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

4.
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

5.
C. Huneke and J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500 crossref(new window)

6.
J. Lipman, On complete ideals in regular local rings, in: Algebraic Geometry and Commutative Algebra, Collected Papers in Honor of Massayoshi Nagata, Academic Press, New York, 1988, pp. 203-231

7.
J. Lipman, Adjoints and polars of simple complete ideals in two-dimensional regular local rings, Bull. Soc. Math. Belgique 45 (1993), 223-244

8.
J. Lipman, Proximity inequalities for complete ideals in two-dimensional regular local rings, Contemporary Math. 159 (1994), 293-306 crossref(new window)

9.
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)

10.
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)

11.
S. Noh, Adjacent integrally closed ideals in dimension two, J. Pure and Applied Algebra 85 (1993), 163-184 crossref(new window)

12.
S. Noh, Powers of simple complete ideals in two-dimensional reular local rings, Comm. Algebra 23 (1995), no. 8, 3127-3143 crossref(new window)

13.
S. Noh, Valuation ideals of order one in two-dimensional regular local rings, Comm. Algebra 28 (2000), no. 2, 613-624 crossref(new window)

14.
S. Noh, Valuation ideals of order two in 2-dimensional regular local rings, Math. Nachr. 261-262 (2003) 123-140 crossref(new window)

15.
P. Ribenboim, The Theory of Classical Valuations, Sringer-Verlag, New York, 1999

16.
O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938) 151-204 crossref(new window)

17.
O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, Princeton, 1960