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The maximum condition on annihilators
for polynomial rings


Authors: Ferran Cedó and Dolors Herbera
Journal: Proc. Amer. Math. Soc. 126 (1998), 2541-2548
MSC (1991): Primary 16P60, 13B25
DOI: https://doi.org/10.1090/S0002-9939-98-04321-4
MathSciNet review: 1451790
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Abstract | References | Similar Articles | Additional Information

Abstract: For each positive integer $n$, we construct a commutative ring ${\mathcal{R}}$ such that the polynomial ring ${\mathcal{R}}[x_{1},\ldots ,x_{n}]$ satisfies the maximum condition on annihilators and ${\mathcal{R}}[x_{1},\ldots ,x_{n+1}]$ does not. In particular, there exists a commutative Kerr ring ${\mathcal{R}}$ such that ${\mathcal{R}}[x]$ is not Kerr. This answers in the negative a question of Faith's.


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

  • 1. V. Camillo and R. Guralnick, Polynomial rings over Goldie rings are often Goldie, Proc. A.M.S. 98 (1986), 567-568. MR 87k:16018
  • 2. C. Faith, Polynomial rings over Goldie-Kerr commutative rings, Proc. A.M.S. 120 (1994), 989-993. MR 94k:13024
  • 3. J. W. Kerr, The polynomial ring over a Goldie ring need not be a Goldie ring, J. Alg. 134 (1990), 344-352. MR 91h:16042
  • 4. M. Roitman, On polynomial extensions of Mori domains over countable fields, J. of Pure and Appl. Algebra 64 (1990), 315-328. MR 91i:13021

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Additional Information

Ferran Cedó
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email: cedo@mat.uab.es

Dolors Herbera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email: dolors@mat.uab.es

DOI: https://doi.org/10.1090/S0002-9939-98-04321-4
Received by editor(s): May 10, 1996
Received by editor(s) in revised form: January 30, 1997
Additional Notes: Both authors are partially supported by the DGICYT (Spain), through the grant PB95-0626, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1998 American Mathematical Society

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