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Ore extensions of Baer and p.p.-rings. (English) Zbl 0982.16021

A ring \(R\) is called Baer (resp. right PP) if the right annihilator of every nonempty subset (resp. singleton subset) is generated as a right ideal by an idempotent. Similarly left PP rings can be defined. A ring is called PP if it is both right and left PP. Let \(\alpha\) be an endomorphism of a ring \(R\). Then according to J. Krempa [Algebra Colloq. 3, No. 4, 289-300 (1996; Zbl 0859.16019)] \(\alpha\) is said to be rigid if \(r\alpha(r)=0\) implies \(r=0\) for \(r\in R\). A ring \(R\) is said to be \(\alpha\)-rigid if there exists a rigid endomorphism \(\alpha\) of \(R\). For an endomorphism \(\alpha\) and \(\delta\) an \(\alpha\)-derivation of a ring \(R\), assume that \(R\) is \(\alpha\)-rigid. Then it is shown that \(R\) is a Baer ring if and only if the Ore extension \(R[x;\alpha,\delta]\) is a Baer ring if and only if the skew power series ring \(R[[x;\alpha]]\) is a Baer ring. Also, it is proved under the same hypothesis that \(R\) is a PP ring if and only if the Ore extension \(R[x;\alpha,\delta]\) is a PP ring.
Reviewer: J.K.Park (Pusan)

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)

Citations:

Zbl 0859.16019
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References:

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