Skew Polynomial Rings with Binomial Relations

Author: Gateva-Ivanova, T.

Source: Journal of Algebra, Volume 185, Number 3, 1 November 1996 , pp. 710-753(44)

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Abstract:

In this paper we continue the study of a class of standard finitely presented quadratic algebras A over a fixed field K , called binomial skew polynomial rings. We consider some combinatorial properties of the set of defining relations F and their implications for the algebraic properties of A . We impose a condition, called (* ), on F and prove that in this case A is a free module of finite rank over a strictly ordered Noetherian domain. We show that an analogue of the Diamond Lemma is true for one-sided ideals of a skew polynomial ring A with condition (* ). We prove, also, that if the set of defining relations F is square free, then condition (* ) is necessary and sufficient for the existence of a finite Groebner basis of every one-sided ideal in A , and for left and right Noetherianness of A . As a corollary we find a class of finitely generated non-commutative semigroups which are left and right Noetherian.

Document Type: Research article

Affiliations: 1: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139

Publication date: 1996-11-01