Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids II

We continue our investigations on algebras R over a field K with generators x1,x2,…,xn subject to (n2) quadratic relations of the form xixj=xkxl with (i,j)≠(k,l) and, moreover, every monomial xixj appears at most once in one of the defining relations. If these relations are non-degenerate then it is shown that the underlying monoid S contains an abelian submonoid A=〈sN|s∈S〉, that is finitely generated and that S=⋃f∈FfA=⋃f∈FAf for some finite subset F of S. So, R=K[S] is a finite module over the Noetherian commutative algebra K[A]; in particular R is a Noetherian algebra that satisfies a polynomial identity. Well-known examples of such monoids are the monoids of I-type that correspond to non-degenerate set-theoretical solutions of the Yang-Baxter equation. We show that S is of I-type if and only if S is cancellative and satisfies the cyclic condition. Furthermore, if S satisfies the cyclic condition, then S is cancellative if and only of K[S] is a prime ring. Moreover, in this case, one can replace the monoid A by a finitely generated submonoid A′ such that fA′=A′f, for each f∈F; in particular R=K[S] is a normalizing extension of K[A′] and thus the prime ideals of K[S] are determined by the prime ideals of K[A′]. These investigations are a continuation and generalization of earlier results of Cedó, Gateva-Ivanova, Jespers and Okniński in the case the defining relations are square free.