Polynomial rings over pseudovaluation rings. (English) Zbl 1167.16021

Bhat, V. K.

Int. J. Math. Math. Sci. 2007, Article ID 20138, 6 p. (2007).

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?

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

Keywords:

skew polynomial rings; pseudovaluation rings; divided rings; strongly prime ideals

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