Skew power series rings over a prime base ring

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 P S 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.