Bimodules in differential polynomial rings

Abstract

We study the $R$-sub-bimodule structure of differential polynomial rings $R[x;δ]$ by introducing the notion of strong simplicity, requiring each nonzero $R$-sub-bimodule of $R[x;δ]$ to be either $R[x;δ]$ or the truncation $\sum_{i=0}^n R x^i$ for some $n \in \mathbb{Z}_{\geq 0}$. Our main result gives a complete characterization: $R[x;δ]$ is strongly simple if and only if $R$ is simple, ${\rm char}(R)=0$, and the derivation $δ$ is outer. We provide examples illustrating both when strong simplicity fails and when it holds.