Let $\mathcal{O}_K$ be a number ring. Until now I thought it was true that $\mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{F}_p^{alg}$ always splits as a finite product $\prod_{i =1}^{n} \mathbb{F}_p^{alg}[\epsilon]/\epsilon^{e_i} \mathbb{F}_p [\epsilon]$. I can't seem to show this though. Could someone show me a proof or a counterexample?
For context, this is relevant to hilbert's theory of ramification of primes in number fields. The ramification indices of $p$ in $\mathcal{O}_K$ occur in the product decomposition as the $e_i$. The degree of $\mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{F}_p^{alg}$ over $\mathbb{F}_p^{alg}$ is the degree of $K$ over $\mathbb{Q}$. $\mathcal{O}_K$ is unramified at $p$ if and only if $\mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{F}_p$ is finite étale over $\mathbb{F}_p$, if and only if $\mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{F}_p^{alg}$ splits as a finite product of $\mathbb{F}_p^{alg}$'s.
It would also be nice to know what sorts of $\mathbb{F}_p$-algebras occur as $\mathcal{O}_K \otimes_{\mathbb{Z}} \mathbb{F}_p$ for some number field $K$, but if there's no class of $\mathbb{F}_p$-algebras that immediately comes to mind, then it could be hard to answer this question.
An $\mathbb{F}_p$-algebra of the form $\mathbb{F}_p[x] / f^e \mathbb{F}_p[x]$ splits after tensoring with $\mathbb{F}_p^{alg}$ into a product $\prod_{i = 1}^{\text{deg}(f)} \mathbb{F}_p^{alg}[\epsilon] / \epsilon^e \mathbb{F}_p^{alg}$.