Non-unital Ore extensions

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta ]$, under the hypothesis that $R$ is $s$-unital and $\ker (\delta )$ contains a non-zero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.