Five solved problems on radicals of Ore extensions
| Autor: | Michał Ziembowski, Be'eri Greenfeld, Agata Smoktunowicz |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Power series
Pure mathematics skew-polynomial extensions General Mathematics Polynomial ring differential polynomial rings Jacobson radical graded nil algebras 01 natural sciences Skew-polynomial extensions 0101 mathematics Mathematics 16N60 16N80 Conjecture 16N20 16N40 010102 general mathematics Graded nil algebras Graded ring Locally nilpotent Skew Differential polynomial rings 16S36 In degree 16S32 16W50 |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Recercat. Dipósit de la Recerca de Catalunya instname Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Publ. Mat. 63, no. 2 (2019), 423-444 Greenfeld, B, Smoktunowicz, A & Ziembowski, M 2019, ' Five solved problems on radicals of Ore extensions ', Publicacions Matemàtiques, vol. 63, no. 2, pp. 423-444 . https://doi.org/10.5565/PUBLMAT6321902 Publicacions Matemàtiques; Vol. 63, Núm. 2 (2019); p. 423-444 |
| ISSN: | 0214-1493 |
| Popis: | The first named author was partially supported by an ISF grant #1623/16. The second named author was supported by ERC Advanced grant Coimbra 320974. The third named author was supported by the Polish National Science Centre grant UMO2017/25/B/ST1/00384. We answer several open questions and establish new results concerningdierential 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 dierential polynomial ring R[X; δ] is locally nilpotent. This answers an open question posed in [41]. The nil radical of a dierential 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 in [30] 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 from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on δ. |
| Databáze: | OpenAIRE |
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