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Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal Av in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of Av. We show that for v ≠ ordR, Av is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(Av; v) is 1. We then give a criterion for R to be regular.

One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal