# Expansivity on Commutative Rings

## TL;DR

This paper extends the concept of expansivity from topological dynamics to automorphisms of commutative rings, characterizing rings with certain expansive automorphisms and exploring their spectral properties.

## Contribution

It introduces the notion of expansivity for ring automorphisms, providing characterizations for rings admitting such automorphisms and analyzing their spectral implications.

## Key findings

- A ring admits a 0-expansive automorphism iff it is a finite product of local rings.
- Rings with positively expansive automorphisms have finitely many maximal ideals.
- The paper explores topological expansivity in the spectrum of rings with the Zariski topology.

## Abstract

In this article we extend the notion of expansivity from topological dynamics to automorphisms of commutative rings with identity. We show that a ring admits a 0-expansive automorphism if and only if it is a finite product of local rings. Generalizing a well known result of compact metric spaces, we prove that if a ring admits a positively expansive automorphism then it admits finitely many maximal ideals. We prove its converse for principal ideal domains. We also consider the topological expansivity induced, in the spectrum of the ring with the Zariski topology, by an automorphism and some consequences are derived.