Skew polynomial rings with binomial relations
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).
Noetherian domains of finite global dimension
scientific article; zbMATH DE number 954780