Bounded skew power series rings for inner $σ$-derivations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916362788536320 |
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| author | Jones, Adam Woods, William |
| author_facet | Jones, Adam Woods, William |
| contents | We define and explore the bounded skew power series ring $R^+[[x;σ,δ]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(σ,δ)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $δ$ is an inner $σ$-derivation and no positive power of $σ$ is inner as an automorphism of $Q$, then $Q^+[[x;σ,δ]]$ is often prime, and even simple under certain mild restrictions on $δ$. It follows from this result that $R^+[[x;σ,δ]]$ is itself prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded skew power series rings for inner $σ$-derivations Jones, Adam Woods, William Rings and Algebras 16S35, 16S36, 16W60, 16W80 We define and explore the bounded skew power series ring $R^+[[x;σ,δ]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(σ,δ)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $δ$ is an inner $σ$-derivation and no positive power of $σ$ is inner as an automorphism of $Q$, then $Q^+[[x;σ,δ]]$ is often prime, and even simple under certain mild restrictions on $δ$. It follows from this result that $R^+[[x;σ,δ]]$ is itself prime. |
| title | Bounded skew power series rings for inner $σ$-derivations |
| topic | Rings and Algebras 16S35, 16S36, 16W60, 16W80 |
| url | https://arxiv.org/abs/2408.10545 |