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In this paper, we investigate the structure of skew power series rings of theform $S = R[[x;\sigma,\delta]]$, where $R$ is a complete filtered ring and$(\sigma,\delta)$ is a skew derivation respecting the filtration. Our mainfocus is on the case in which $\sigma\delta = \delta\sigma$, and we aim to usetechniques in non-commutative valuation theory to address the long-standingopen question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime idealof $S$? When $R$ has characteristic $p$, our results reduce this to afinite-index problem. We also give preliminary results in the "Iwasawa algebra"case $\delta = \sigma - \mathrm{id}_R$ in arbitrary characteristic. A key stepin 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.

Skew power series rings over a prime base ring