Proper divisibility in computable rings

Abstract

We study divisibility in computable integral domains. We develop a technique for coding Σ20 binary trees into the divisibility relation of a computable integral domain. We then use this technique to prove two theorems about non-atomic integral domains.In every atomic integral domain, the divisibility relation is well-founded. We show that this classical theorem is equivalent to ACA0 over RCA0.In every computable non-atomic integral domain there is a Δ30 infinite sequence of proper divisions. We show that this upper bound cannot be improved to Δ20 in general.