Five solved problems on radicals of Ore extensions

Be′eri Greenfeld , Agata Smoktunowicz , Michał Ziembowski

Abstract

We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In par- ticular, 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 [41]. 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 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.
Author Be′eri Greenfeld
Be′eri Greenfeld,,
-
, Agata Smoktunowicz
Agata Smoktunowicz,,
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, Michał Ziembowski (FMIS / DAC)
Michał Ziembowski,,
- Department of Algebra and Combinatorics
Journal seriesPublicacions Matematiques, ISSN 0214-1493, (A 35 pkt)
Issue year2019
Vol63
No2
Pages423-444
Publication size in sheets1.05
Keywords in Polishrozszerzenia skośnych wielomianów, pierścienie wielomianów z różniczkowaniem, radykał Jacobsona, zgradowane nil algebry
Keywords in EnglishSkew-polynomial extensions, differential polynomial rings, Jacobson radical, graded nil algebras
ASJC Classification2600 General Mathematics
Abstract in PolishW pracy podano odpowiedź na pięć problemów związanych z rozszerzeniami pierścieni nieprzemiennych do skośnych wielomianów
DOIDOI:10.5565/PUBLMAT6321902
URL http://mat.uab.es/pubmat/volums/navegador
Languageen angielski
Score (nominal)35
ScoreMinisterial score = 35.0, 20-08-2019, ArticleFromJournal
Publication indicators Scopus SNIP (Source Normalised Impact per Paper): 2016 = 0.672; WoS Impact Factor: 2017 = 1.0 (2) - 2017=1.019 (5)
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