Invariants associated with ideals in one-dimensional local domains

Let R be a one-dimensional local Noetherian domain with maximal ideal m, quotient field K and residue field R/m:=k. We assume that the integral closure of R in its quotient field K is a DVR and a finite R-module. We assume also that the field k is isomorphic to the residue field of . For I a proper ideal of R, denote the inverse of I by I∗; that is, I∗ is the set (R:KI) of elements of K that multiply I into R. We investigate two numerical invariants associated to a proper ideal I of R that have previously come up in the literature from various points of view. The two invariants are: (1) the difference between the composition lengths of I∗/R and R/I, and (2) the difference between the product, when the composition length of R/I is multiplied by the composition length of m∗/R, and the length of I∗/R. We show that these two differences can be expressed in terms of the type sequence of R, a finite sequence of positive integers related to the natural valuation inherited from .