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a b s t r a c tComplete ideals adjacent to the maximal ideal of a two-dimensional regular local ring (called first neighborhood complete ideals) have been studied by S. Noh.Here these ideals are studied in the more general case of a two-dimensional Muhly local domain, i.e., an integrally closed Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.It is shown that certain properties of such a local ring R (e.g. being regular or being a rational singularity in case the embedding dimension is 3) can be characterized in terms of the first neighborhood complete ideals.A necessary and sufficient condition for an immediate prime divisor of R to possess an asymptotically irreducible ideal is derived. If, in addition, R is a rational singularity, we are able to give a description of such an ideal.

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We give a particularly short and elementary proof of the length formula of Hoskin and Deligne. Unlike Johnston and Verma (Math. Proc. Cambridge Philos. Soc. 111, 423-432 (1992)), we don't first prove a formula for the multiplicity of a complete m-primary ideal of a two-dimensional regular local ring (R, m).

Let (R, M) be a two-dimensional Muhly local domain, i.e., an integrally closed Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain. Motivated by recent work of Ciuperca, Heinzer, Ratliff and Rush on projectively full ideals, we prove that every complete ideal adjacent to the maximal ideal M is projectively full.