Skew power series rings over a prime base ring

Abstract

In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;\sigma,\delta]]$, where $R$ is a complete filtered ring and $(\sigma,\delta)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $\sigma\delta = \delta\sigma$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $\delta = \sigma - \mathrm{id}_R$ in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is "almost" $(\sigma,\delta)$-invariant in a certain sense.