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On nilpotent elements of Ore extensions. (English) Zbl 1383.16034

Summary: Let \(R\) be an associative ring with unity, {\(\alpha\)} be an endomorphism of \(R\) and {\(\delta\)} an {\(\alpha\)}-derivation of \(R\). We introduce the notion of {\(\alpha\)}-nilpotent p.p.-rings, and prove that the {\(\alpha\)}-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when \(R\) is a nil-{\(\alpha\)}-compatible and 2-primal ring with Nil(\(R\)) nilpotent, then Nil(\(R[x;\alpha,\delta])=\mathrm{Nil}(R)[x;\alpha,\delta]\); and when \(R\) is a nil Armendriz ring of skew power series type with Nil(\(R\)) nilpotent, then \(\mathrm{Nil}(R[[x;\alpha]])=\mathrm{Nil}(R)[[x;\alpha]]\), where Nil(R) is the set of nilpotent elements of \(R\). These results extend existing results to a more general setting.

MSC:

16U20 Ore rings, multiplicative sets, Ore localization
16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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