Every local ring is dominated

by a one-dimensional local ring

Authors:
Robert Gilmer and William Heinzer

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2513-2520

MSC (1991):
Primary 13B02, 13C15, 13E05, 13H99

MathSciNet review:
1389520

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[AM]**M. F. Atiyah and I. G. Macdonald,*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802****[CHL]**Paul-Jean Cahen, Evan G. Houston, and Thomas G. Lucas,*Discrete valuation overrings of Noetherian domains*, Proc. Amer. Math. Soc.**124**(1996), no. 6, 1719–1721. MR**1317033**, 10.1090/S0002-9939-96-03260-1**[Ch]**C. Chevalley,*La notion d’anneau de décomposition*, Nagoya Math. J.**7**(1954), 21–33 (French). MR**0067866****[C]**I. S. Cohen,*On the structure and ideal theory of complete local rings*, Trans. Amer. Math. Soc.**59**(1946), 54–106. MR**0016094**, 10.1090/S0002-9947-1946-0016094-3**[DL]**Ada Maria de Souza Doering and Yves Lequain,*The gluing of maximal ideals—spectrum of a Noetherian ring—going up and going down in polynomial rings*, Trans. Amer. Math. Soc.**260**(1980), no. 2, 583–593. MR**574801**, 10.1090/S0002-9947-1980-0574801-5**[G]**Robert Gilmer,*Multiplicative ideal theory*, Queen’s Papers in Pure and Applied Mathematics, vol. 90, Queen’s University, Kingston, ON, 1992. Corrected reprint of the 1972 edition. MR**1204267****[GH1]**Robert Gilmer and William Heinzer,*The Noetherian property for quotient rings of infinite polynomial rings*, Proc. Amer. Math. Soc.**76**(1979), no. 1, 1–7. MR**534377**, 10.1090/S0002-9939-1979-0534377-2**[GH2]**Robert Gilmer and William Heinzer,*Ideals contracted from a Noetherian extension ring*, J. Pure Appl. Algebra**24**(1982), no. 2, 123–144. MR**651840**, 10.1016/0022-4049(82)90009-3**[GH3]**Robert Gilmer and William Heinzer,*The family of residue fields of a zero-dimensional commutative ring*, J. Pure Appl. Algebra**82**(1992), no. 2, 131–153. MR**1182935**, 10.1016/0022-4049(92)90117-X**[GH4]**Robert Gilmer and William Heinzer,*Artinian subrings of a commutative ring*, Trans. Amer. Math. Soc.**336**(1993), no. 1, 295–310. MR**1102887**, 10.1090/S0002-9947-1993-1102887-7**[GH5]**Robert Gilmer and William Heinzer,*Imbeddability of a commutative ring in a finite-dimensional ring*, Manuscripta Math.**84**(1994), no. 3-4, 401–414. MR**1291129**, 10.1007/BF02567465**[HL]**William Heinzer and David Lantz,*Ideal theory in two-dimensional regular local domains and birational extensions*, Comm. Algebra**23**(1995), no. 8, 2863–2880. MR**1332150**, 10.1080/00927879508825373**[H]**James A. Huckaba,*Commutative rings with zero divisors*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 117, Marcel Dekker, Inc., New York, 1988. MR**938741****[M]**Hideyuki Matsumura,*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273****[N]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
13B02,
13C15,
13E05,
13H99

Retrieve articles in all journals with MSC (1991): 13B02, 13C15, 13E05, 13H99

Additional Information

**Robert Gilmer**

Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027

Email:
gilmer@math.fsu.edu

**William Heinzer**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395

Email:
heinzer@math.purdue.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-03847-1

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