Algebraic and Turing separability of rings
type: Item
instance of: scholarly article
title: Algebraic and Turing separability of rings
zbMATH Open document ID: 0860.12006
author: Alexandra Shlapentokh
published in: Journal of Algebra
publication date: 1997-04-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 12L05
Mathematics Subject Classification ID: 03D45
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 03D25
zbMATH DE Number: 941234
zbMATH Keywords: Turing separability of rings
zbMATH Keywords: weak presentation of field rings
zbMATH Keywords: recursively enumerable Turing degrees
zbMATH Keywords: total recursive functions
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
full work available at URL: jabr.1996.0323
OpenAlex ID: W2035752609
DOI: 10.1006/JABR.1996.0323
description: scientific article; zbMATH DE number 941234