Revisiting virtual difference ideals

TL;DR

This paper corrects a previous error regarding virtual ideals in difference algebra, demonstrating their properties and conditions under which they simplify to periodic ideals, thereby enhancing the theoretical framework.

Contribution

It rectifies an earlier mistake in the theory of virtual difference ideals and clarifies when these ideals coincide with periodic ideals.

Findings

Corrected the theoretical framework for virtual ideals

Identified conditions where virtual ideals are equivalent to periodic ideals

Extended the applicability of difference algebra concepts

Abstract

The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic ideals alone, we had to generalise further to a somewhat ephemeral setting called virtual ideals. This text has two purposes. It corrects an error in [4] discovered by Tom Scanlon's UCB seminar, recovering all results for all virtual ideals. In addition, based on results in [3], we describe a wide family of difference equations where virtual ideals reduce to periodic ideals.