Every local ring is dominated by a onedimensional local ring
Authors:
Robert Gilmer and William Heinzer
Journal:
Proc. Amer. Math. Soc. 125 (1997), 25132520
MSC (1991):
Primary 13B02, 13C15, 13E05, 13H99
MathSciNet review:
1389520
Fulltext PDF Free Access
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Abstract: Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated algebra.
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 [CHL]
 P.J. Cahen, E. G. Houston, and T. G. Lucas, Discrete valuation overrings of Noetherian domains, Proc. Amer. Math. Soc. 124 (1996), 17191721. MR 96h:13057
 [Ch]
 C. Chevalley, La notion d'anneau de de\'{c}omposition, Nagoya Math. J. 7 (1954), 2133. MR 16:788g
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 [DL]
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 [G]
 R. Gilmer, Multiplicative Ideal Theory, Queen's Papers Pure Appl. Math. Vol 90, Kingston, 1992. MR 93j:13001
 [GH1]
 R. Gilmer and W. Heinzer, The Noetherian property for quotient rings of infinite polynomial rings, Proc. Amer. Math. Soc. 76 (1979), 17. MR 80h:13010
 [GH2]
 R. Gilmer and W. Heinzer, Ideals contracted from a Noetherian extension ring, J. Pure Appl. Algebra 24 (1982), 123144. MR 84a:13006
 [GH3]
 R. Gilmer and W. Heinzer, The family of residue fields of a zerodimensional commutative ring, J. Pure Appl. Algebra 82 (1992), 131153. MR 93k:13019
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 R. Gilmer and W. Heinzer, Artinian subrings of a commutative ring, Trans. Amer. Math. Soc. 336 (1993), 295310. MR 93e:13028
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 R. Gilmer and W. Heinzer, Imbeddability of a commutative ring in a finitedimensional ring, Manuscripta Math. 84 (1994), 401414. MR 95i:13004
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 J. Huckaba, Commutative Rings with Zero Divisors, MarcelDekker, New York, 1988. MR 89e:13001
 [M]
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 [N]
 M. Nagata, Local Rings, Interscience, 1962. MR 27:5790
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Additional Information
Robert Gilmer
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 323063027
Email:
gilmer@math.fsu.edu
William Heinzer
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 479071395
Email:
heinzer@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002993997038471
PII:
S 00029939(97)038471
Received by editor(s):
August 4, 1995
Received by editor(s) in revised form:
March 12, 1996
Communicated by:
Wolmer Vasconcelos
Article copyright:
© Copyright 1997 American Mathematical Society
