Quasi-one-fibered ideals in two-dimensional Muhly local domains

In this paper, a two-dimensional integrally closed Noetherian local domain (R,M) with algebraically closed residue field is said to be a Muhly local domain, if the associated graded ring is an integrally closed domain.Using the theory of degree functions developed by Rees and Sharp, and the Zariski–Lipman theory of complete ideals in two-dimensional regular local rings, we study M-primary ideals of R having among their Rees valuations exactly one valuation ≠ordR. We therefore call these ideals quasi-one-fibered. Some consequences are deduced.