Recognition: no theorem link
On a q-Skew Amitsur's Theorem
Pith reviewed 2026-05-12 01:23 UTC · model grok-4.3
The pith
The constant part of the Jacobson radical of q-skew Ore extensions is nil
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Let R be an algebra over an uncountable field, σ a locally torsion automorphism and δ a locally nilpotent left σ-derivation such that qσδ = δσ, where q is a nonzero scalar. The constant part of the Jacobson radical of the Ore extension R[x;σ,δ] is nil. This partially answers a question of Greenfeld, Smoktunowicz and Ziembowski posed in 2019. As a corollary, employing Shin's 2024 result, the Jacobson radical of R[x;σ, δ] is N[x;σ,δ] for some nil ideal N of R when the field has characteristic zero.
What carries the argument
The Ore extension R[x;σ,δ] equipped with the local conditions on σ and δ and the q-commutation relation, which is used to prove nilpotency of constant radical elements.
If this is right
- The constant terms of Jacobson radical elements in R[x;σ,δ] form a nil ideal in R.
- In characteristic zero the Jacobson radical equals N[x;σ,δ] for a nil ideal N of R.
- This provides a partial solution to the structure of radicals in skew polynomial rings.
- Amitsur-type theorems extend to q-skew cases under these hypotheses.
Where Pith is reading between the lines
- The result might hold without the uncountable field assumption for many common algebras over countable fields.
- The proof techniques could extend to other skew extensions or quantum polynomial rings.
- Explicit checks on examples such as differential operator rings or Weyl algebras would test whether the nil constant property persists.
Load-bearing premise
The base field is uncountable, σ is locally torsion, and δ is locally nilpotent while commuting with σ up to the scalar q.
What would settle it
An example of such an Ore extension over a countable field or with non-locally-torsion σ where a non-nilpotent constant lies in the Jacobson radical would disprove the claim.
read the original abstract
Let $R$ be an algebra over an uncountable field, $\sigma$ a locally torsion automorphism and $\delta$ a locally nilpotent left $\sigma$-derivation such that $q\sigma\delta = \delta\sigma$, where $q$ is a nonzero scalar. We show that the constant part of the Jacobson radical of the Ore extension $R[x;\sigma,\delta]$ is nil. This partially answers a question of Greenfeld, Smoktunowicz and Ziembowski posed in 2019. As a corollary, we employ Shin's 2024 result to prove a q-skew Amitsur's theorem whenever the field is additionally assumed to be of characteristic zero. That is, the Jacobson radical of $R[x;\sigma, \delta]$ is $N[x;\sigma,\delta]$ for some nil ideal $N$ of $R$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that if R is an algebra over an uncountable field, σ is a locally torsion automorphism, and δ is a locally nilpotent left σ-derivation satisfying qσδ = δσ for nonzero scalar q, then the constant part of the Jacobson radical of the Ore extension R[x;σ,δ] is nil. As a corollary, when the base field has characteristic zero, Shin's 2024 theorem yields that the full Jacobson radical equals N[x;σ,δ] for some nil ideal N of R. This partially answers a 2019 question of Greenfeld, Smoktunowicz and Ziembowski.
Significance. If correct, the result supplies a partial q-skew analogue of Amitsur's theorem on radicals of polynomial rings, under local hypotheses on σ and δ rather than global ones. The argument proceeds by iterated use of the given commutation relation together with local nilpotency and torsion to control supports (via uncountability) and establish nilpotency of constant elements in the radical. The corollary then invokes an external 2024 result to obtain the full radical description in characteristic zero. This contributes to the theory of Jacobson radicals in Ore extensions and may facilitate further work on skew versions of classical radical theorems.
minor comments (3)
- §2 (proof of the main theorem): the iteration that produces higher powers of a constant element in the radical is described only schematically; an explicit inductive step with the precise exponent arising from local nilpotency of δ would improve readability.
- Introduction, paragraph 3: the statement that the result 'partially answers' the 2019 question would benefit from a one-sentence indication of which hypotheses from the original question remain open.
- Notation section: the symbol for the constant part of the Jacobson radical is introduced without an explicit definition; adding a displayed equation would prevent ambiguity with the usual constant term map.
Simulated Author's Rebuttal
We thank the referee for the positive report, accurate summary of the main theorem, and recommendation for minor revision. The assessment of significance is appreciated, particularly the connection to the 2019 question of Greenfeld, Smoktunowicz and Ziembowski and the partial q-skew analogue of Amitsur's theorem.
Circularity Check
No significant circularity; derivation self-contained via standard Ore-extension arguments and external citation
full rationale
The paper proves that the constant part of the Jacobson radical of R[x;σ,δ] is nil by iterated application of the given relation qσδ=δσ together with the local nilpotency of δ and local torsion of σ, using uncountability only to control supports in the standard manner for such results. This relies on the definitions of Ore extensions and the Jacobson radical without any reduction of the claim to a fitted quantity, self-definition, or self-citation chain. The corollary applies Shin's independent 2024 theorem under the additional characteristic-zero hypothesis. No load-bearing step reduces to the paper's inputs by construction, and the cited result is externally falsifiable.
Axiom & Free-Parameter Ledger
axioms (3)
- standard math Jacobson radical is the intersection of all maximal left ideals
- standard math Ore extension R[x;σ,δ] is defined by the usual twisted multiplication rules
- domain assumption Uncountable base field
Reference graph
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