Injective modules and linear growth of primary decompositions
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- by R. Y. Sharp PDF
- Proc. Amer. Math. Soc. 128 (2000), 717-722 Request permission
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.References
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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
- Received by editor(s): May 5, 1998
- Published electronically: October 6, 1999
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 717-722
- MSC (1991): Primary 13E05
- DOI: https://doi.org/10.1090/S0002-9939-99-05170-9
- MathSciNet review: 1641105