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

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

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 . 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.

**[A]**J. Apel,*A relationship between Gröbner bases of ideals and vector modules of**-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**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.

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

Email:
xkramer@member.ams.org

DOI:
https://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