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**Polynomial rings over pseudovaluation rings.**
*(English)*
Zbl 1167.16021

Let \(R\) be a ring with unit element, and let \(\sigma\) be an automorphism of \(R\). Then \(R\) is called a pseudo-valuation ring (PVR) if every prime ideal \(P\) of \(R\) is strongly prime, that is, if \(aP\subseteq bR\) or \(bR\subseteq aP\) for all \(a,b\in R\); \(R\) is called \(\sigma\)-divided (divided) if every prime ideal of \(R\) is comparable, under inclusion, to every \(\sigma\)-stable ideal (ideal).

Suppose that \(x\not\in P\) for every prime ideal \(P\) of the skew polynomial ring \(R[x,\sigma]\). Then \(R[x,\sigma]\) is a PVR or \(\sigma\)-divided whenever \(R\) is a commutative PVR or \(\sigma\)-divided, respectively. Similar results hold for the Ore extension \(R[x,\delta]\) of a commutative Noetherian \(\mathbb{Q}\)-algebra \(R\) with derivation \(\delta\): if \(R\) is a PVR or divided, then so is \(R[x,\delta]\). The paper concludes with an open question: If \(R\) is a commutative PVR with an automorphism \(\sigma\) and a \(\sigma\)-derivation \(\delta\), does it follow that \(R[x,\sigma,\delta]\) is a PVR?

Suppose that \(x\not\in P\) for every prime ideal \(P\) of the skew polynomial ring \(R[x,\sigma]\). Then \(R[x,\sigma]\) is a PVR or \(\sigma\)-divided whenever \(R\) is a commutative PVR or \(\sigma\)-divided, respectively. Similar results hold for the Ore extension \(R[x,\delta]\) of a commutative Noetherian \(\mathbb{Q}\)-algebra \(R\) with derivation \(\delta\): if \(R\) is a PVR or divided, then so is \(R[x,\delta]\). The paper concludes with an open question: If \(R\) is a commutative PVR with an automorphism \(\sigma\) and a \(\sigma\)-derivation \(\delta\), does it follow that \(R[x,\sigma,\delta]\) is a PVR?

Reviewer: Günter Krause (Winnipeg)

### MSC:

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

13A15 | Ideals and multiplicative ideal theory in commutative rings |

16D25 | Ideals in associative algebras |

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