# On monoids of ideals of orders in quadratic number fields

## TL;DR

This paper investigates the algebraic structure of ideals in orders of quadratic number fields, specifically analyzing their factorization properties such as catenary degrees and sets of lengths.

## Contribution

It provides a comprehensive determination of the catenary degrees, distances, and unions of sets of lengths for ideals in quadratic orders, advancing understanding of their factorization behavior.

## Key findings

- Determined the set of catenary degrees for ideals in quadratic orders.
- Identified the set of distances and unions of sets of lengths.
- Enhanced the understanding of non-unique factorization in algebraic number theory.

## Abstract

We determine the set of catenary degrees, the set of distances, and the unions of sets of lengths of the monoid of nonzero ideals and of the monoid of invertible ideals of orders in quadratic number fields.