LENGTH FUNCTIONS ON INTEGRAL-DOMAINS

Let R be an integral domain and x is-an-element-of 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, l(x)BAR and L(x)BAR, of l(x(n))/n and L(x(n))/n, respectively, as n goes to infinity always exist. Moreover, for any 0 less-than-or-equal-to alpha less-than-or-equal-to 1 less-than-or-equal-to beta less-than-or-equal-to infinity, there is an integral domain R and an irreducible x is-an-element-of R such that l(x)BAR = alpha and L(x)BAR = beta.