Long length functions

Abstract

Let D be an integral domain satisfying ACCP. We refine the classical notion of (factorization) length by recursively defining the length of a nonzero element to be the least ordinal strictly greater than the lengths of its proper divisors. This gives a surjective function L:D⁎→L(D), where L(D), called the length of D, is the least ordinal strictly greater than the length of any nonzero element. We show that an ordinal is the length of a domain satisfying ACCP if and only if it is of the form ωβ. We give some conditions for when monoid domains, generalized power series domains, inert extensions, or localizations at splitting sets satisfy ACCP, and calculate the lengths of these domains in these cases. Finally, for each positive integer n≥2 and each ordinal μ≥n, we construct a domain D satisfying ACCP and an x∈D⁎ with L(x)=μ and l(x)=n, where l(x) denotes the number of factors in a minimum length atomic factorization of x.