Bounded skew power series rings for inner $\sigma$-derivations

Abstract

We define and explore the bounded skew power series ring $R^{+}[[x;\sigma ,\delta ]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(\sigma ,\delta )$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $\delta$ is an inner $\sigma$-derivation and no positive power of $\sigma$ is inner as an automorphism of $Q$, then $Q^{+}[[x;\sigma ,\delta ]]$ is often prime, and even simple under certain mild restrictions on $\delta$. It follows from this result that $R^{+}[[x;\sigma ,\delta ]]$ is itself prime.