1. A completion of $\mathbb{Z}$ is a field
- Creator:
- Marcos, José E.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- sequential convergence, convergence ring, and completion of a convergence ring
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public