Skew Polynomial Rings with Binomial Relations

Abstract

In this paper we continue the study of a class of standard finitely presented quadratic al¬gebras 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 al¬gebraic 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.