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 uniquel ...

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 anm-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.

Ⅰ. INTRODUCTION
Ⅱ. Valuation ideals of small rank
Ⅲ. b = odd case
Ⅳ. b = even case
Ⅴ. Sequence of v-ideals
REFERENCES

Ⅰ. INTRODUCTION
Ⅱ. Valuation ideals of small rank
Ⅲ. b = odd case
Ⅳ. b = even case
Ⅴ. Sequence of v-ideals
REFERENCES