Ideals contracted from 1-dimensional

overrings with an application

to the primary decomposition of ideals

Authors:
William Heinzer and Irena Swanson

Journal:
Proc. Amer. Math. Soc. **125** (1997), 387-392

MSC (1991):
Primary 13C05, 13E05, 13H99

DOI:
https://doi.org/10.1090/S0002-9939-97-03703-9

MathSciNet review:
1363423

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that each ideal of a locally formally equidimensional analytically unramified Noetherian integral domain is the contraction of an ideal of a one-dimensional semilocal birational extension domain. We give an application to a problem concerning the primary decomposition of powers of ideals in Noetherian rings. It is shown in an earlier paper by the second author that for each ideal in a Noetherian commutative ring there exists a positive integer such that, for all , there exists a primary decomposition where each contains the -th power of its radical. We give an alternate proof of this result in the special case where is locally at each prime ideal formally equidimensional and analytically unramified.

**[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****[B]**M. Brodmann,*Asymptotic stability of 𝐴𝑠𝑠(𝑀/𝐼ⁿ𝑀)*, Proc. Amer. Math. Soc.**74**(1979), no. 1, 16–18. MR**521865**, https://doi.org/10.1090/S0002-9939-1979-0521865-8**[GH]**Robert Gilmer and William Heinzer,*Ideals contracted from a Noetherian extension ring*, J. Pure Appl. Algebra**24**(1982), no. 2, 123–144. MR**651840**, https://doi.org/10.1016/0022-4049(82)90009-3**[HO]**William Heinzer and Jack Ohm,*Noetherian intersections of integral domains*, Trans. Amer. Math. Soc.**167**(1972), 291–308. MR**0296095**, https://doi.org/10.1090/S0002-9947-1972-0296095-6**[He]**Jürgen Herzog,*A homological approach to symbolic powers*, Commutative algebra (Salvador, 1988) Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 32–46. MR**1068322**, https://doi.org/10.1007/BFb0085535**[Hu]**Craig Huneke,*Uniform bounds in Noetherian rings*, Invent. Math.**107**(1992), no. 1, 203–223. MR**1135470**, https://doi.org/10.1007/BF01231887**[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****[Rat]**L. J. Ratliff Jr.,*On prime divisors of 𝐼ⁿ, 𝑛 large*, Michigan Math. J.**23**(1976), no. 4, 337–352 (1977). MR**0457421****[R1]**D. Rees,*A note on analytically unramified local rings*, J. London Math. Soc.**36**(1961), 24–28. MR**0126465**, https://doi.org/10.1112/jlms/s1-36.1.24**[R2]**D. Rees,*A note on asymptotically unmixed ideals*, Math. Proc. Cambridge Philos. Soc.**98**(1985), no. 1, 33–35. MR**789716**, https://doi.org/10.1017/S0305004100063210**[S1]**Irena Swanson,*Primary decompositions of powers of ideals*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 367–371. MR**1266193**, https://doi.org/10.1090/conm/159/01518**[S2]**-,*Powers of Ideals: Primary decompositions, Artin-Rees lemma and regularity,*, Math. Annalen (to appear).**[ZS]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. 1*, Springer-Verlag, New York-Heidelberg-Berlin, 1975. With the cooperation of I. S. Cohen; Corrected reprinting of the 1958 edition; Graduate Texts in Mathematics, No. 28. MR**0384768**

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Additional Information

**William Heinzer**

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

Email:
heinzer@math.purdue.edu

**Irena Swanson**

Affiliation:
Department of Mathematics, New Mexico State University, Las Cruces, New Mexico 88003-8001

Email:
iswanson@nmsu.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03703-9

Received by editor(s):
September 6, 1995

Additional Notes:
The authors thank Craig Huneke for helpful suggestions concerning this paper.

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1997
American Mathematical Society