Norms, Normsets, and Factorization

TL;DR

This paper explores the concept of norms and normsets, examining their algebraic properties and relationships to factorization, with implications for multiplicative ideal theory and domain analysis.

Contribution

It introduces and analyzes the notion of normsets as monoids with unique factorization properties, extending the understanding of norms in algebraic structures.

Findings

Normsets are monoids with specific factorization properties.

Relationship established between factoring in domains and their normsets.

Potential applications in multiplicative ideal theory.

Abstract

We present a development of norms and discuss their relationship to factorization. In earlier work, the first named author introduced the notion of a normset, which is the image of the norm map. A normset is a monoid with its own factorization properties. We discuss in several different environments the relationship between factoring in domains and their respective normsets. We will also discuss the utility of this notion when it comes to multiplicative ideal theory.