Algebraic and Turing Separability of Rings

Abstract

LetFbe a countable field/ring. Then aweak presentationofFis an injective homomorphism fromFinto a field/ring whose universe is2such that all the field/ring operations are translated by total recursive functions. Given two recursive integral domainsR1andR2with quotient fieldsF1andF2respectively, we investigate under what circumstances there exists a weak presentation of the fieldF1F2such that the images ofR1andR2belong to two different recursively enumerable (r.e.) Turing degrees. In many cases we succeed in giving a completely algebraic necessary and sufficient condition for the “Turing separation” described above. More specifically, under some conditions, we can make the images ofR1andR2be of arbitrary r.e. degrees. The algebraic condition is a generalization of the notion of algebraic field separability. As a result of our investigation, we also show that3has an r.e. weak presentation as a ring which is not a weak presentation of4as a field.