A completion of ℤ is a field

José E. Marcos

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 3, page 689-706
  • ISSN: 0011-4642

Abstract

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We define various ring sequential convergences on ℤ and ℚ . We describe their properties and properties of their convergence completions. In particular, we define a convergence 𝕃 1 on 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 ℤ / ( p ) . Further, we show that ( ℤ , 𝕃 1 * ) is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

How to cite

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Marcos, José E.. "A completion of $\mathbb {Z}$ is a field." Czechoslovak Mathematical Journal 53.3 (2003): 689-706. <http://eudml.org/doc/30809>.

@article{Marcos2003,
abstract = {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.},
author = {Marcos, José E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {sequential convergence; convergence ring; completion of a convergence ring; sequential convergence; convergence ring; completion of a convergence ring},
language = {eng},
number = {3},
pages = {689-706},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A completion of $\mathbb \{Z\}$ is a field},
url = {http://eudml.org/doc/30809},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Marcos, José E.
TI - A completion of $\mathbb {Z}$ is a field
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 3
SP - 689
EP - 706
AB - 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.
LA - eng
KW - sequential convergence; convergence ring; completion of a convergence ring; sequential convergence; convergence ring; completion of a convergence ring
UR - http://eudml.org/doc/30809
ER -

References

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