Adjacent integrally closed ideals in 2-dimensional regular local rings

Let be a 2-dimensional regular local ring with algebraically closed residue field. Zariskiʹs Unique Factorization Theorem asserts that every integrally closed (complete) -primary ideal I is uniquely factored into a product of powers of simple complete ideals , where Pi is a simple complete ideal for ai 1 and n 1. In this paper, we give a new characterization for a simple complete ideal in terms of adjacent complete ideals. We also give a characterization for a complete ideal I to have finitely many adjacent complete -primary over-ideals. Namely, we show that I is simple if and only if it has a unique adjacent over-ideal and that has only finitely many complete adjacent over-ideals if and only if ai=1 for every i and there are no proximity relations among Pi.