The Noetherian property in some quadratic algebras

Abstract

Abstract. We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations-namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2-and on the Ufnarovskii graph Γ(A). The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph Γ(A) to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.

Quadratic algebras of skew type and the underlying monoids

“…Notice that, under this additional assumption on the relations, every element of S can be written in the form x k1 1 • • • x kn n for some non-negative integers k i . In particular, the Gelfand-Kirillov dimension of K[S], denoted by GK(K[S]), does not exceed n. We note that some other (but related) types of algebras defined by quadratic relations have been investigated, see for example [5], [6], [12].…”

Constructing Noetherian Algebras

“…Recently, in the contexts of Gr' obner basis, set theoretic solutions of the Yang-Baxter equation, and subsemigroups of polycyclic-by-finite groups (see for example [8,12,13,14,15,27,33]), the Noetherian property has been investigated for finitely generated algebras of the type K (Xl) X2) ... ) Xn R), with R a set of relations of the type u = v, where u and v are words in the generators Xl) X2) ..• ) Xn and K a field. Of course, such an algebra coincides with the semigroup algebra K[ S] of the monoid S defined by the same presentation.…”