Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-99-05170-9
Published electronically: October 6, 1999
MathSciNet review: 1641105
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • [He-S] W. Heinzer and I. Swanson, Ideals contracted from $1$-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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13E05

Retrieve articles in all journals with MSC (1991): 13E05


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

American Mathematical Society