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Algebraic and Turing Separability of Rings

✍ Scribed by Alexandra Shlapentokh

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 311 KB
Volume: 185
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.0323

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✦ Synopsis

Let F be a countable fieldrring. Then a weak presentation of F is an injective homomorphism from F into a fieldrring whose universe is ‫ގ‬ such that all the fieldrring operations are translated by total recursive functions. Given two recursive integral domains R and R with quotient fields F and F respectively, we

investigate under what circumstances there exists a weak presentation of the field F F such that the images of R and R belong to two different recursively 1 2 1 2

Ž . 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 of R 1 and R be of arbitrary r.e. degrees. The algebraic condition is a generalization of 2 the notion of algebraic field separability. As a result of our investigation, we also show that ‫ޑ‬ has an r.e. weak presentation as a ring which is not a weak presentation of ‫ޑ‬ as a field.

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