On value sets of fractional ideals

TL;DR

This paper explores the duality of fractional ideals in admissible rings, characterizing canonical ideals through symmetry relations in their value sets, extending known results in algebraic geometry.

Contribution

It introduces new symmetry characterizations of canonical ideals and extends the understanding of value set relationships for dual fractional ideals.

Findings

Characterization of canonical ideals via symmetry relations.

Extension of symmetry among maximal points in value sets.

Generalization of previous results on dual ideals.

Abstract

The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings of algebraic curves at singular points. We characterize canonical ideals by means of a symmetry relation between lengths of certain quotients of associated ideals to a pair of dual ideals. In particular, we extend the symmetry among absolute and relative maximals in the sets of values of pairs of dual fractional ideals to other kinds of maximal points. Our results generalize and complement previous ones by other authors.