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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 $I$ in a Noetherian commutative ring $R$ there exists a positive integer $k$ such that, for all $n \ge 1$, there exists a primary decomposition $I^{n} = Q_{1} \cap \dots \cap Q_{s}$ where each $Q_{i}$ contains the $nk$-th power of its radical. We give an alternate proof of this result in the special case where $R$ is locally at each prime ideal formally equidimensional and analytically unramified.


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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

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