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Skew polynomial rings satisfying \(r\)-\(Bnd\) property. (English) Zbl 0818.16020
All rings have identity and all modules are unitary. A ring \(R\) is said to satisfy \(r\)-\(T(R) = n\) if every finitely generated torsion (with respect to the regular elements of \(R\)) right \(R\)-module can be generated by \(n\) elements. The ring \(R\) is said to satisfy \(r\)-\(Bnd(n)\) if every right ideal can be generated by \(n\) elements. Consider the Ore extension \(R[x,\sigma,\delta]\), where \(\sigma\) is an automorphism and \(\delta\) a \(\sigma\)-derivation of \(R\). It is proved that if \(R\) is prime right Noetherian with \(r\)-\(T(R) = n\) then \(R[x,\sigma,\delta]\) satisfies \(r\)- \(Bnd(n+1)\). If \(R\) is simple right Noetherian with right Krull dimension \(n\) then both \(R\) and \(R[x,\sigma,\delta]\) satisfy \(r\)-\(Bnd(n+1)\). Conversely, if \(R\) is prime right Noetherian and \(R[x,\sigma]\) satisfies \(r\)-\(Bnd(n)\) then \(R\) has \(r\)-\(T(R) = n\). For related results see the papers of J. T. Stafford [Bull. Lond. Math. Soc. 8, 168-173 (1976; Zbl 0327.16010); Commun. Algebra 8, 1513-1518 (1980; Zbl 0436.16010)].
MSC:
16P40 Noetherian rings and modules (associative rings and algebras)
16D25 Ideals in associative algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
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