The model theory of residue rings of models of Peano Arithmetic: The prime power case

Abstract

In \cite{MacResField} the second author gave a systematic analysis of definability and decidability for rings $\mathcal{M}/p\mathcal{M}$, where $\mathcal{M}$ is a model of Peano Arithmetic and $p$ is a prime in $\mathcal{M}$. In the present paper we extend those results to the more difficult case of $\mathcal{M}/p^k\mathcal{M}$, where $\mathcal{M}$ is a model of Peano Arithmetic, $p$ is a prime in $\mathcal{M}$, and $k>1$. In \cite{MacResField} work of Ax on finite fields was used, here we use in addition work of Ax on ultraproduct of $p$-adics.