Skew Polynomial Rings with Binomial Relations

Abstract

In this paper we continue the study of a class of standard finitely presented quadratic algebrasAover a fixed fieldK, called binomial skew polynomial rings. We consider some combinatorial properties of the set of defining relationsFand their implications for the algebraic properties ofA. We impose a condition, called (∗), onFand prove that in this caseAis 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 ringAwith condition (∗). We prove, also, that if the set of defining relationsFis square free, then condition (∗) is necessary and sufficient for the existence of a finite Groebner basis of every one-sided ideal inA, and for left and right Noetherianness ofA. As a corollary we find a class of finitely generated non-commutative semigroups which are left and right Noetherian.