Multiplicative Relationships of Subrings and their Applications to Factorization

TL;DR

This paper introduces new ways to compare subrings within larger rings, focusing on their properties and applications to factorization, especially in the context of orders in number fields.

Contribution

It defines three comparison methods for subrings and explores their implications, providing new insights into the structure of orders in number fields and their factorization properties.

Findings

An order in a number field is associated iff it is ideal-preserving and locally associated.

The properties of being associated, ideal-preserving, and locally associated are characterized and related.

Applications to identifying and constructing orders in number fields are discussed.

Abstract

When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being associated, being ideal-preserving, or being locally associated. We then explore how these properties of a subring might be leveraged to give information about $R$, including applications to the field of factorization. Of particular interest is the result that an order in a number field is associated if and only if it is both ideal-preserving and locally associated. We conclude with a discussion of how these properties are realized in the case of orders in a number field and how such orders might be found.