Skew polynomial rings with binomial relations
type: Item
instance of: scholarly article
title: Skew polynomial rings with binomial relations
zbMATH Open document ID: 0863.16016
author: Tatiana Gateva-Ivanova
published in: Journal of Algebra
publication date: 1997-06-09T00:00:00Z
full work available at URL: http://hdl.handle.net/10525/3336
review text: The paper is concerned with algebras \(A\) over a field \(K\) defined by generators \(x_1,\dots,x_n\) subject to relations of the form \(x_jx_i=c_{ij}x_kx_l\), \(1\leq i<j\leq n\) where \(c_{ij}\in K\setminus\{0\}\) and \(k\leq l\), \(k<j\). It is also assumed that these relations yield a reduced Gröbner basis of the ideal of the free algebra \(K\langle x_1,\dots,x_n\rangle\) defining \(A\) and, for the main results, that the following condition is satisfied: each pair \(k\), \(l\) with \(k<l\) appears in the right hand side of a defining relation. Combinatorial properties of algebras of this type (and of the corresponding semigroups obtained by replacing all \(c_{ij}\) by 1), referred to as binomial skew polynomial rings, are investigated. In particular, existence of Gröbner bases of one sided ideals is established. It is also shown that \(A\) is Noetherian. This is a continuation of the author's previous article [Trans. Am. Math. Soc. 343, No. 1, 203-219 (1994; Zbl 0807.16026)]. Reviewer's note: In a forthcoming paper [Semigroups of I-type (to appear)] of the author and \textit{M. Van den Bergh}, combinatorial properties of a more general class of semigroups are further studied and related to other algebraic notions of general interest. In particular, such algebras are shown to be Noetherian domains of finite global dimension. The structural properties of binomial algebras and of the underlying semigroups are treated in ``Binomial semigroups'' by \textit{E. Jespers} and the reviewer (to appear).
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 954780
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: reduced Gröbner bases
zbMATH Keywords: free algebras
zbMATH Keywords: binomial skew polynomial rings
zbMATH Keywords: semigroups
zbMATH Keywords: Noetherian domains of finite global dimension
zbMATH Keywords: binomial algebras
reviewed by: Jan Okniński
MaRDI profile type: Publication
OpenAlex ID: W2023115398
DOI: 10.1006/JABR.1996.0348
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description: scientific article; zbMATH DE number 954780