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 $S=R\left[\right[x;\sigma ,\delta \left]\right]$, where R is a complete, positively 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.