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Five solved problems on radicals of ore extensions
https://ddd.uab.cat/record/206878
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. Greenfeld, Be'eriTue, 02 Jul 2019 04:44:32 GMThttps://ddd.uab.cat/record/2068782019Duo, Bézout and distributive rings of skew power series
https://ddd.uab.cat/record/49668
We give necessary and sufficient conditions on a ring R and an endomorphism σ of R for the skew power series ring R[[x; σ]] to be right duo right Bézout. In particular, we prove that R[[x; σ]] is right duo right Bézout if and only if R[[x; σ]] is reduced right distributive if and only if R[[x; σ]] is right duo of weak dimension less than or equal to 1 if and only if R is N0-injective strongly regular and σ is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative B'ezout power series rings. Mazurek, RyszardFri, 16 Oct 2009 13:11:07 GMThttps://ddd.uab.cat/record/496682009