Lezama, Oswaldo; Acosta, Juan Pablo; Chaparro, Cristian; Ojeda, Ingrid; Venegas, César Ore and Goldie theorems for skew PBW extensions. (English) Zbl 1291.16018 Asian-Eur. J. Math. 6, No. 4, Article ID 1350061, 20 p. (2013). 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. Reviewer: Vyacheslav A. Artamonov (Moskva) Cited in 8 Documents 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 Keywords:skew polynomial extensions; localizations; Ore theorem; Goldie theorem; Poincaré-Birkhoff-Witt extensions; iterated polynomial extensions; left Ore domains; semiprime left Goldie rings PDFBibTeX XMLCite \textit{O. Lezama} et al., Asian-Eur. J. Math. 6, No. 4, Article ID 1350061, 20 p. (2013; Zbl 1291.16018) Full Text: DOI arXiv References: [1] DOI: 10.1006/jabr.1994.1336 · Zbl 0820.17015 · doi:10.1006/jabr.1994.1336 [2] DOI: 10.1070/RM1998v053n04ABEH000056 · Zbl 0931.16002 · doi:10.1070/RM1998v053n04ABEH000056 [3] DOI: 10.2140/pjm.1988.131.13 · Zbl 0598.16002 · doi:10.2140/pjm.1988.131.13 [4] DOI: 10.1007/978-94-017-0285-0 · doi:10.1007/978-94-017-0285-0 [5] DOI: 10.1016/S0021-8693(02)00542-2 · Zbl 1017.16017 · doi:10.1016/S0021-8693(02)00542-2 [6] DOI: 10.1017/CBO9780511542794 · doi:10.1017/CBO9780511542794 [7] Gelfand I., Publ. Math. Inst. Hautes Études. Sci. 31 pp 509– [8] DOI: 10.1017/CBO9780511841699 · doi:10.1017/CBO9780511841699 [9] DOI: 10.1090/S0002-9939-1974-0379617-3 · doi:10.1090/S0002-9939-1974-0379617-3 [10] DOI: 10.1007/s10468-005-0707-y · Zbl 1090.16011 · doi:10.1007/s10468-005-0707-y [11] Lezama O., Comm. Algebra 39 pp 50– [12] DOI: 10.1090/gsm/030 · doi:10.1090/gsm/030 [13] DOI: 10.2307/1968173 · Zbl 0007.15101 · doi:10.2307/1968173 [14] DOI: 10.1006/jabr.2000.8463 · Zbl 0966.17008 · doi:10.1006/jabr.2000.8463 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.