Bounded and finite factorization 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.