[Paper Review] On Radicals of Ore Extensions and Related Questions

Be’eri Greenfeld, Agata Smoktunowicz+1 more|arXiv (Cornell University)|Feb 26, 2017

Advanced Topics in Algebra30 references3 citations

TL;DR

This paper resolves several open problems in noncommutative ring theory concerning radicals in Ore extensions and differential polynomial rings. It establishes that if R is a prime radical ring with a derivation δ, then the differential polynomial ring R[X;δ] is locally nilpotent, and proves that the nil radical of R[X;δ] is of the form I[X;δ] when the base field is infinite—answering long-standing questions by Nielsen, Ziembowski, and Hong et al. Additionally, it shows Jacobson radical stability under grading-preserving derivations in characteristic zero, and confirms the well-definedness of power series rings with locally nilpotent σ-derivations, generalizing prior results.

ABSTRACT

We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Nielsen for algebras over infinite fields. If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $δ$ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $δ$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring $R[[X;σ,δ]]$ is well-defined whenever $δ$ is a locally nilpotent $σ$-derivation; this answers a conjecture by Bergen and Grzeszczuk and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $δ$.

Motivation & Objective

• To resolve open questions regarding the structure of radicals in differential and skew polynomial rings.
• To investigate the behavior of nil and Jacobson radicals under derivations in Ore extensions.
• To determine conditions under which differential polynomial rings are locally nilpotent or have radicals of a specific form.
• To establish the well-definedness of power series rings with locally nilpotent σ-derivations, extending prior work.

Proposed method

• Use of ideal-theoretic techniques to analyze the structure of radicals in differential polynomial rings R[X;δ].
• Application of properties of prime radical rings and locally nilpotent derivations to prove local nilpotency of R[X;δ].
• Leveraging the assumption of an infinite base field to characterize the nil radical as I[X;δ] for some ideal I of R.
• Employing grading and characteristic zero assumptions to prove δ-stability of the Jacobson radical in graded algebras.
• Analysis of skew polynomial rings with natural grading to show equivalence between local and graded local nilpotency.
• Proof that power series rings R[[X;σ,δ]] are well-defined when δ is a locally nilpotent σ-derivation, confirming a conjecture by Bergen and Grzeszczuk.

Experimental results

Research questions

• RQ1 Under what conditions is the differential polynomial ring R[X;δ] locally nilpotent when R is prime radical?
• RQ2 When is the nil radical of R[X;δ] of the form I[X;δ] for some ideal I of R, particularly over infinite fields?
• RQ3 Is the Jacobson radical δ-stable under a grading-preserving derivation δ in a graded algebra over a field of characteristic zero?
• RQ4 Are skew polynomial rings with natural grading locally nilpotent if and only if they are graded locally nilpotent?
• RQ5 Is the power series ring R[[X;σ,δ]] well-defined when δ is a locally nilpotent σ-derivation?

Key findings

• If R is prime radical and δ is a derivation on R, then the differential polynomial ring R[X;δ] is locally nilpotent, resolving an open question by Nielsen and Ziembowski.
• When the base field is infinite, the nil radical of R[X;δ] is of the form I[X;δ] for some ideal I of R, answering an open question by Hong, Kim, Lee, and Nielsen.
• For a graded algebra R generated in degree 1 over a field of characteristic zero, and a grading-preserving derivation δ, the Jacobson radical of R is δ-stable, and this result is sharp as shown by counterexamples.
• Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent, establishing a precise equivalence.
• The power series ring R[[X;σ,δ]] is well-defined whenever δ is a locally nilpotent σ-derivation, confirming a conjecture by Bergen and Grzeszczuk and enabling broader generalizations in the theory of Ore extensions.