Work

Skew power series rings over a prime base ring

Year: 2024

Type: article

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 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 $δ= σ- \mathrm{id}_R$ 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" $(σ,δ)$-invariant in a certain sense.

Source: Journal of Pure and Applied Algebra

DOI: 10.1016/j.jpaa.2024.107800

Authors: Adam Jones, William Woods

Language: English

Topic: Rings, Modules, and Algebras

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,330

Funders: Engineering and Physical Sciences Research Council, Heilbronn Institute for Mathematical Research

Award: G7192499587