Injective modules and linear growth

of primary decompositions

Author:
R. Y. Sharp

Journal:
Proc. Amer. Math. Soc. **128** (2000), 717-722

MSC (1991):
Primary 13E05

DOI:
https://doi.org/10.1090/S0002-9939-99-05170-9

Published electronically:
October 6, 1999

MathSciNet review:
1641105

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Abstract | References | Similar Articles | Additional Information

Abstract: The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

**[He-S]**W. Heinzer and I. Swanson,*Ideals contracted from -dimensional overrings with an application to the primary decomposition of ideals*, Proc. Amer. Math. Soc.**125**(1997), 387-392. MR**97d:13008****[H-H]**M. Hochster and C. Huneke,*Tight closure, invariant theory, and the Briançon-Skoda Theorem*, J. Amer. Math. Soc.**3**(1990), 31-116. MR**91g:13010****[Hu]**C. Huneke,*Uniform bounds in Noetherian rings*, Invent. Math.**107**(1992), 203-223. MR**93b:13027****[K]**D. Kirby,*Artinian modules and Hilbert polynomials*, Quart. J. Math. Oxford (2)**24**(1973), 47-57. MR**47:4993****[M]**H. Matsumura,*Commutative ring theory*, Cambridge University Press, Cambridge, 1986. MR**88h:13001****[S-S]**K. E. Smith and I. Swanson,*Linear bounds on growth of associated primes*, Communications in Algebra**25**(1997), 3071-3079. MR**98k:13003****[S]**I. Swanson,*Powers of ideals: primary decompositions, Artin-Rees lemma and regularity*, Math. Annalen**307**(1997), 299-313. MR**97j:13005**

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Additional Information

**R. Y. Sharp**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Email:
r.y.sharp@sheffield.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-99-05170-9

Keywords:
Commutative Noetherian ring,
primary decomposition,
associated prime ideal,
injective module,
Artin-Rees Lemma

Received by editor(s):
May 5, 1998

Published electronically:
October 6, 1999

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1999
American Mathematical Society