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A completion of $\mathbb{Z}$ is a field

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…