Every local ring is dominated by a one-dimensional local ring

Gilmer, Robert; Heinzer, William

doi:10.1090/S0002-9939-97-03847-1

Every local ring is dominated by a one-dimensional local ring
HTML articles powered by AMS MathViewer

by Robert Gilmer and William Heinzer PDF

Proc. Amer. Math. Soc. 125 (1997), 2513-2520 Request permission

Abstract:

Let $(R, \mathbf {m})$ be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring $S$ of $R$ such that (i) $S$ dominates $R$, (ii) the residue field of $S$ is a finite purely transcendental extension of $R/ \mathbf {m}$, (iii) every associated prime of (0) in $S$ contracts in $R$ to an associated prime of (0), and (iv) $\dim (S) \le 1$. In addition, it is shown that $S$ can be obtained so that either $\mathbf {m} S$ is the maximal ideal of $S$ or $S$ is a localization of a finitely generated $R$-algebra.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
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, DOI 10.1090/S0002-9939-96-03260-1
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
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, DOI 10.1090/S0002-9947-1980-0574801-5
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
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, DOI 10.1090/S0002-9939-1979-0534377-2
Robert Gilmer and William Heinzer, Ideals contracted from a Noetherian extension ring, J. Pure Appl. Algebra 24 (1982), no. 2, 123–144. MR 651840, DOI 10.1016/0022-4049(82)90009-3
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, DOI 10.1016/0022-4049(92)90117-X
Robert Gilmer and William Heinzer, Artinian subrings of a commutative ring, Trans. Amer. Math. Soc. 336 (1993), no. 1, 295–310. MR 1102887, DOI 10.1090/S0002-9947-1993-1102887-7
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, DOI 10.1007/BF02567465
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, DOI 10.1080/00927879508825373
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
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
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856