Algebraic and Turing separability of rings
A weak presentation of a countable field/ring \(F\) is an injective homomorphism from \(F\) into a field/ring whose universe is \(\mathbb{N}\) such that all the field/ring operations are translated by total recursive functions. Given two recursive integral domains \(R_1\) and \(R_2\) with quotient field \(F_1\) and \(F_2\) respectively, in this paper the author investigates under what circumstances there exists a weak presentation of the field \(F_1F_2\) such that the images of \(R_1\) and \(R_2\) belong to two different recursively enumerable (r.e.) Turing degrees. Finally the author show that \(\mathbb{Q}\) has an r.e. weak presentation as a ring which is not a weak presentation of \(\mathbb{Q}\) as a field.
scientific article; zbMATH DE number 941234