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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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 $\mathfrak{a}$ in a commutative Noetherian ring $R$ has linear growth of primary decompositions, that is, there exists a positive integer $h$ such that, for every positive integer $n$, there exists a minimal primary decomposition ${\mathfrak{a}}^{n} = {\mathfrak{q}}_{n1} \cap \ldots \cap {\mathfrak{q}}_{nk_{n}}$ with $\sqrt {{\mathfrak{q}}_{ni}}^{hn} \subseteq {\mathfrak{q}}_{ni}$ for all $i =1, \ldots , k_{n}$. The generalization involves a finitely generated $R$-module and several ideals; the short proof is based on the theory of injective $R$-modules.


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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: http://dx.doi.org/10.1090/S0002-9939-99-05170-9
PII: S 0002-9939(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