INTEGRALLY CLOSED IDEALS AND REES VALUATION

Let (R, m, k) be a normal Noetherian analytically irreducible universally catenary local domain of dimension d >= 1 with infinite residue field k and field of fractions K, and let I be an m-primary ideal. For P is an element of Min(mR[It]) there exists at least one Q is an element of Min(mR[It]) such that P = Q boolean AND R[It]. Let v be the Rees valuation of V := R[It](Q) boolean AND K. We obtain necessary and sufficient conditions for the quotient ring R[It]/P to be a polynomial ring in d variables over k in terms of the v-values of a suitably chosen minimal generating set for I. Let J := (a(1,) ... , a(d))R be a minimal reduction of I. If I is a normal ideal and R [It]/P is a polynomial ring in d variables over k, we show that the residue field k(v) of V, is generated over k by the images of a(1)/a(d), ... , a(d-1)/a(d). If (R, m, k) is a two-dimensional regular local ring with algebraically closed residue field k and I is a product of distinct simple integrally closed m-primary ideals, we show for each positive integer n and each Q is an element of Min(mR[I(n)t]) that the ring R[I(n)t]/Q is a two-dimensional normal Cohen-Macaulay graded domain with minimal multiplicity at its maximal homogeneous ideal, with this multiplicity being n.