The Noetherian property in some quadratic algebras

Author:
Xenia H. Kramer

Journal:
Trans. Amer. Math. Soc. **352** (2000), 4295-4323

MSC (2000):
Primary 16P40; Secondary 16S15, 16S37

Published electronically:
May 23, 2000

MathSciNet review:
1665334

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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 . The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph 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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Additional Information

**Xenia H. Kramer**

Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003

Address at time of publication:
Department of Mathematics and Computer Science, Dickinson College, Carlisle, Pennsylvania 17013

Email:
xkramer@member.ams.org

DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02493-4

Keywords:
Noncommutative Noetherian ring,
standard finitely presented algebra,
noncommutative Gr\"{o}bner basis,
quadratic algebra

Received by editor(s):
October 8, 1997

Published electronically:
May 23, 2000

Additional Notes:
This paper was written as partial fulfillment of the requirements for the Ph.D. degree at New Mexico State University under the direction of R. Laubenbacher, who has the author’s warmest gratitude.

Article copyright:
© Copyright 2000
American Mathematical Society