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

[摘要] We continue our investigations on algebras R over a field K with generators x(1), x(2),... x(n) subject to (n 2) quadratic relations of the form x(i)x(j) = x(k)x(l) with (i, j) not equal (k, l) and, moreover, every monomial x(i)x(j) 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 = < s(N) | s is an element of S >, that is finitely generated and that S = U-f is an element of F f A =U(f is an element of F)Af 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 is an element of 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 Cedo, Gateva-Ivanova, Jespers and Okninski in the case the defining relations are square free. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Noetherian algebra;Finitely presented;Quadratic relations;Semigroup algebra