The Noetherian property in some quadratic algebras |
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Authors: | Xenia H Kramer |
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Institution: | Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003 |
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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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Keywords: | Noncommutative Noetherian ring standard finitely presented algebra noncommutative Gr\"{o}bner basis quadratic algebra |
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