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A remark on the Jacobson property of PI Ore extensions. (Une remarque sur la propriété de Jacobson des extensions de Ore a I.P.) (French) Zbl 0819.16024
The Jacobson condition is investigated for Ore extensions (skew polynomial rings) of the form \(R[X;\delta]\) where \(R\) is a noetherian polynomial identity ring and \(\delta\) is a derivation on \(R\). Let \(\mathcal I\) be the set of those \(\delta\)-prime ideals of \(R\) such that \(R/I\) has finite uniform rank (Goldie dimension) over its subring of central \(\delta\)-constants. The author proves that \(R[X;\delta]\) is a Jacobson ring if and only if \(R/I\) is Jacobson for all \(I \in {\mathcal I}\). The case when \(R\) is commutative was proved by R. B. Warfield jun. and the reviewer [Math. Z. 180, 503-523 (1982; Zbl 0495.16002)]. Necessary and sufficient conditions for \(R[X;\delta]\) to be primitive were given in the same paper for \(R\) commutative, and extended to the case when \(R\) is P.I. by the present author [in Commun. Algebra 20, 1819-1837 (1992; Zbl 0754.16014)].
MSC:
16S32 Rings of differential operators (associative algebraic aspects)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16S36 Ordinary and skew polynomial rings and semigroup rings
16P50 Localization and associative Noetherian rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16W25 Derivations, actions of Lie algebras
16U20 Ore rings, multiplicative sets, Ore localization
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[2] K. R. Goodearl andR. B. Warfield Jr., Primitivity in differential operator rings. Math. Z.180, 503-523 (1982). · Zbl 0495.16002 · doi:10.1007/BF01214722
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[4] A. Ouarit, Extensions de Ore d’anneaux noethériens à I.P. Comm. Algebra20, 1819-1837 (1992). · Zbl 0754.16014 · doi:10.1080/00927879208824433
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