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;σ,δ]], where R is a complete, positively filtered ring and (σ,δ) is a skew derivation respecting the filtration. Our main focus is on the case in which σδ=δσ, 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 δ=σ−idR 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” (σ,δ)-invariant in a certain sense.