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Valuation ideals of order two in 2-dimensional regular local rings

Title
Valuation ideals of order two in 2-dimensional regular local rings
Authors
Noh S.
Ewha Authors
노선숙
SCOPUS Author ID
노선숙scopus
Issue Date
2003
Journal Title
Mathematische Nachrichten
ISSN
0025-584XJCR Link
Citation
vol. 261-262, pp. 123 - 140
Indexed
SCI; SCIE; SCOPUS WOS scopus
Abstract
Let K be the quotient field of a 2-dimensional regular local ring (R, m) and let v be a prime divisor of R, i.e., a valuation of K birationally dominating R which is residually transcendental over R. Zariski showed that: such prime divisor v is uniquely associated to a simple m-primary integrally closed ideal I of R, there are only finitely many simple v-ideals including I, and all the other v-ideals can be uniquely factored into products of simple v-ideals. The number of nonmaximal simple v-ideals is called the rank of v or the rank of I as well. It is also known that such an m-primary ideal I is minimally generated by o(I) +1 elements, where o(I) denotes the m-adic order of I. Given a simple valuation ideal of order two associated to a prime divisor v of arbitrary rank, in this paper we find minimal generating sets of all the simple v-ideals and the value semigroup v(R) in terms of its rank and the v-value difference of two elements in a regular system of parameters of R. We also obtain unique factorizations of all the composite v-ideals and describe the complete sequence of v-ideals as explicitly as possible. © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
DOI
10.1002/mana.200310116
Appears in Collections:
사범대학 > 수학교육과 > Journal papers
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