The study of the ideals in a regular local ring (R,m) of dimension 2 has a long and important tradition dating back to the fundamental work of Zariski [ZS]. More recent contributions are due to several authors including Cutkosky, Huneke, Lipman, Rees, Sally, and Teissier among others, see [C1,C2,H,HS,L,LT,R]. One of the main result in this setting is the unique factorization theorem for complete (i.e., integrally closed) ideals proved originally by Zariski [ZS, Theorem 3, Appendix 5]. It asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way (up to the order of the factors). By definition, an ideal is simple if it cannot be written as a product of two proper ideals. Another important property of a complete ideal I is that its reduction number is 1 which in turns implies that the associated graded ring grI (R) is Cohen–Macaulay and its Hilbert series is well-understood; this is due to Lipman and Teissier [LT], see also [HS]. The class of contracted ideals plays an important role in the original work of Zariski as well as in the work of Huneke [H]. An ideal I of R is contracted if I = R ∩ IR[m/x] for some x ∈ m \m2. Any complete ideal is contracted but not the other way round. The associated graded ring grI (R) of a contracted ideal I need not be Cohen–Macaulay and its Hilbert series can be very complicated.