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Prime spectra of ambiskew polynomial rings. (English) Zbl 1445.16024

Summary: We determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra \(R\) over an algebraically closed field \(\mathbb{K}\) to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra \(U(sl_2)\) (in characteristic 0) and its quantization \(U_q(sl_2)\) (when \(q\) is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of \(R\) to consist of 0, the ideals \((z-\lambda)R\) for some central element \(z\) of \(R\) and all \(\lambda\in\mathbb{K}\), and, for some positive integer \(d\) and each positive integer \(m\), \(d\) height two prime ideals \(P\) for which \(R/P\) has Goldie rank \(m\).

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16D30 Infinite-dimensional simple rings (except as in 16Kxx)
16N60 Prime and semiprime associative rings
16W20 Automorphisms and endomorphisms
16W25 Derivations, actions of Lie algebras
16U20 Ore rings, multiplicative sets, Ore localization
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