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In \cite{MacResField} the second author gave a systematic analysis ofdefinability 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 onfinite fields was used, here we use in addition work of Ax on ultraproduct of$p$-adics.

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