Generalised Stability of ultraproducts of finite residue rings

Abstract

We study ultraproducts of finite residue rings $\underset{n\in\mathbb{N}}{\prod}\mathbb{Z}/n\mathbb{Z} \diagup \mathcal{U} $ where $\mathcal{U}$ is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter $\mathcal{U}$ to determine if the resulting ultraproduct $\underset{n\in\mathbb{N}}{\prod}\mathbb{Z}/n\mathbb{Z} \diagup \mathcal{U}$ has simple, NIP, NTP2 but not simple nor NIP, or TP2 theory, noting that all these four cases occur.