Bounded and finite factorization domains

TL;DR

This paper surveys key results on bounded and finite factorization domains, which are classes of atomic integral domains with bounded or finitely many factorizations, respectively, highlighting their properties and significance in algebra.

Contribution

It provides a comprehensive survey of important findings related to bounded and finite factorization domains, clarifying their definitions and roles in algebraic factorization theory.

Findings

Bounded factorization domains have a uniform bound on the length of factorizations.

Finite factorization domains have finitely many factorizations for each element.

The survey summarizes foundational and recent results in the study of these domains.

Abstract

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive integer $N$ such that for any factorization $x = a_1 \cdots a_n$ of $x$ into irreducibles $a_1, \dots, a_n$ in $R$, the inequality $n \le N$ holds. In addition, we say that $R$ is a finite factorization domain if it is atomic and every nonzero nonunit in $R$ factors into irreducibles in only finitely many ways (up to order and associates). The notions of bounded and finite factorization domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in their systematic study of factorization in atomic integral domains. Here we provide a survey of some of the most relevant results on bounded and finite factorization domains.