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Title: A completion of $\mathbb{Z}$ is a field (English)
Author: Marcos, J. E.
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 53
Issue: 3
Year: 2003
Pages: 689-706
Summary lang: English
Category: math
Summary: We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence $\mathbb{L}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/(p)$. Further, we show that $(\mathbb{Z}, \mathbb{L}^\ast _1)$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan. (English)
Keyword: sequential convergence
Keyword: convergence ring
Keyword: completion of a convergence ring
MSC: 13J10
MSC: 13J99
MSC: 54A20
MSC: 54H13
idZBL: Zbl 1080.54500
idMR: MR2000063
Date available: 2009-09-24T11:05:46Z
Last updated: 2020-07-03
Stable URL:
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