  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.
