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Existence of prime elements in rings of generalized power series

Published online by Cambridge University Press:  12 March 2014

Daniel Pitteloud
Daniel Pitteloud*
Affiliation:
Université de Lausanne, Institute D'Informatique, 1015 Lausanne, Switzerland, E-mail: daniel.pitteloud@math.unige.ch
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Abstract

The field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved that K((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove that K((G≤0)) does have prime elements if G = (ℝ, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1206 - 1216
Copyright
Copyright © Association for Symbolic Logic 2001

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References

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