Bimodules in differential polynomial rings

Abstract

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

7 pages