Quotient rings of integers from a metric point of view

Abstract

The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers. It is shown the existence of limit rings (fields) as the Gromov-Hausdorff limits of sequences of metric quotient rings. The relation of these constructions with the field of the reals $\mathbb{R}$ is discussed, showing that they are dense in $\mathbb{R}$ but that they cannot be identified with the real field or with the rational field $\mathbb{Q}$, at least when $\mathbb{R}$ and $\mathbb{Q}$ are endowed with the usual metric structures. It is also shown that the limit rings can be endowed with an order relation.