Length Functions on Integral Domains

Abstract

Let $R$ be an integral domain and $x \in R$ which is a product of irreducible elements. Let $l(x)$ and $L(x)$ denote respectively the inf and sup of the lengths of factorizations of $x$ into a product of irreducible elements. We show that the two limits, $\bar l(x)$ and $\bar L(x)$, of $l({x^n})/n$ and $L({x^n})/n$, respectively, as $n$ goes to infinity always exist. Moreover, for any $0 \leq \alpha \leq 1 \leq \beta \leq \infty$, there is an integral domain $R$ and an irreducible $x \in R$ such that $\bar l(x) = \alpha$ and $\overline L (x) = \beta$.