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

Abstract

Let v be a prime divisor of a 2-dimensional regular local ring (R, m) with algebraically closed residue field k. Zariski showed that: a prime divisor v of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple v-ideals including I, and all the other v-ideals can be uniquely factored into products of simple v-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements. Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor v, we find minimal generating sets of all the simple v-ideals and describe factorizations of all the composite v-ideals in terms of power products of simple v-ideals as explicitly as possible. Copyright © 2000 by Marcel Dekker, Inc.