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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References:

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