Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [A] J. Apel, A relationship between Gröbner bases of ideals and vector modules of $G$-algebras, Comtemporary Mathematics, vol. 131 (Part 2), Amer. Math. Soc., Providence, Rhode Island, 1992, pp. 195-204. MR 94f:68102
  • [AS] M. Artin and W. Schelter, Graded algebras of global dimension 3, Advances in Mathematics 66 (1987), 171-216. MR 88k:16003
  • [ATV] M. Artin, J. Tate, and M. Van Den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Progress in Mathematics, vol. 86, Birkhäuser, Boston, 1990, pp. 33-85. MR 92e:14002
  • [B] G. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978), 178-218. MR 81b:16001
  • [Bu] B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, Ph.D. Thesis, Univ. Innsbruck (Austria), 1965.
  • [D] L. Dickson, Finiteness of the odd perfect and primitive abundant numbers with $n$ distinct prime factors, Amer. J. Math. 35 (1913), 413-422.
  • [GI1] T. Gateva-Ivanova, On the Noetherianity of some associative finitely presented algebras, J. Algebra 138 (1991), 13-35. MR 92g:16031
  • [GI2] -, Noetherian properties and growth of some associative algebras, Effective Methods in Algebraic Geometry, Progress in Mathematics, vol. 94, Birkhäuser, Boston, 1991, pp. 143-158. MR 92c:16025
  • [GI3] -, Noetherian properties of skew polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994), 203-219. MR 94g:16029
  • [GI4] -, Skew Polynomial Rings with Binomial Relations, J. Algebra 185 (1996), 710-753. MR 97m:16047
  • [KRW] A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Computation 9 (1990), 1-26. MR 91e:13025
  • [KL] G. Krause and T. Lenagan, Growth of algebras and the Gelfand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman Publishing Inc., Marshfield, Massachusetts, 1985. MR 86g:16001
  • [MR] J. McConnell and J. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, New York, 1987. MR 89j:16023
  • [M1] F. Mora, Groebner bases for non-commutative polynomial rings, Proc. AAECC-3, Lecture Notes in Computer Science, vol. 229, 1986, pp. 353-362. CMP 19:04
  • [M2] T. Mora, Groebner bases in non-commutative algebras, Proc. ISSAC 88, Lecture Notes in Computer Science, vol. 358, 1988, pp. 150-161. MR 90k:68082
  • [M3] -, Seven variations on standard bases, preprint, Università di Genova, Dipartimento di Matematica, 1988.
  • [M4] -, An introduction to commutative and noncommutative Gröbner bases, Theoretical Computer Science 134 (1994), 131-173. MR 95i:13027
  • [O] J. Okninski, On monomial algebras, Archiv der Mathematik 50 (1988), 417-423. MR 89d:16036
  • [SmSt] S. Smith and J. Stafford, Regularity of the four dimensional Sklyanin algebra, Compositio Mathematica 83 (1992), 259-289. MR 93h:16037
  • [U1] V. Ufnarovskii, A growth criterion for graphs and algebras defined by words, Math. Notes 31 (1982), 238-241.
  • [U2] -, On the use of graphs for computing a basis, growth and Hilbert series of associative algebras, Math. USSR, Sb. 68 (2) (1991), 417-428.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16P40, 16S15, 16S37

Retrieve articles in all journals with MSC (2000): 16P40, 16S15, 16S37

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

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

American Mathematical Society