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

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)