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Ore and Goldie theorems for skew PBW extensions. (English) Zbl 1291.16018

Let \(R\) be a subring of an associative ring \(A\) such that \(A\) is generated over \(R\) by elements \(x_1,\ldots,x_n\) with defining relations \[ x_ir-c_{i,r}x_i\in R,\quad x_jx_i -c_{i,j}x_ix_j\in R+Rx_1+\cdots+Rx_n \] where \(c_{i,j},c_{i,r}\in R\setminus\{0\}\). It is assumed that \(A\) is a free left \(R\)-module with a base consisting of monomials \(x_1^{m_1}\cdots x_n^{m_n}\). The extension \(R\subset A\) is a skew PBW-extension.
If \(A\) admits a grading with respect to monomials in \(x_1,\ldots,x_n\), then \(A\) is quasi-commutative. It is an iterated polynomial extension of \(R\) with automorphisms. Suppose that \(R\) is a left Ore domain. Then so is \(A\) and \(A\) has a total division ring of fractions. If \(R\) is a semiprime left Goldie ring, so is \(A\).
A quantum analog of the Gelfand-Kirillov conjecture is proved for skew quantum polynomials.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16U20 Ore rings, multiplicative sets, Ore localization
16T20 Ring-theoretic aspects of quantum groups
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