The Noetherian property in some quadratic algebras

Kramer, Xenia

doi:10.1090/S0002-9947-00-02493-4

The Noetherian property in some quadratic algebras
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by Xenia H. Kramer PDF

Trans. Amer. Math. Soc. 352 (2000), 4295-4323 Request permission

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 $\Gamma (A)$. The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph $\Gamma (A)$ 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.

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