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

Fish, Christopher D.; Jordan, David A.

Glasg. Math. J. 61, No. 1, 49-68 (2019).

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

Keywords:

ambiskew polynomial rings; Goldie rank

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