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 \R 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 \R . For I a proper ideal of R, denote the inverse of I by I*; that is, I* is the set (R:_K I) 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 \R.

### Invariants associated with ideals in one-dimensional local domains

#### Abstract

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 \R 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 \R . For I a proper ideal of R, denote the inverse of I by I*; that is, I* is the set (R:_K I) 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 \R.
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2007
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11567/251286`
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