Five solved problems on radicals of ore extensions
Greenfeld, Be'eri (Bar Ilan University (Israel). Department of Mathematics)
Smoktunowicz, Agata (University of Edinburgh. School of Mathematics)
Ziembowski, Michal (Warsaw University of Technology (Polònia). Faculty of Mathematics and Information Science)

Data:	2019
Resum:	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.
Nota:	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.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Skew-polynomial extensions ; Differential polynomial rings ; Jacobson radical ; Graded nil algebras
Publicat a:	Publicacions matemàtiques, Vol. 63 Núm. 2 (2019) , p. 423-444 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/358948
DOI: 10.5565/PUBLMAT6321902

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