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On asymptotic values of certain sets of attached prime ideals

Published online by Cambridge University Press:  18 May 2009

A.-J. Taherizadeh
A.-J. Taherizadeh
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH
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In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 3 , September 1988 , pp. 293 - 300
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Brodmann, M., Asymptotic stability of Ass(M/InM), Proc. Amer. Math. Soc. 74 (1979), 16–18.Google Scholar
2.Kirby, D., Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford Ser. (2) 24 (1973), 47–57.CrossRefGoogle Scholar
3.Kirby, D., Coprimary decomposition of Artinian modules, J. London Math. Soc. (2) 6 (1973), 571–576.CrossRefGoogle Scholar
4.Macdonald, I. G., Secondary representation of modules over a commutative ring, Symposia Mathematica 11 (1973), 23–43.Google Scholar
5.McAdam, S., Asymptotic prime divisors, Lecture Notes in Mathematics 1023 (Springer, 1983).CrossRefGoogle Scholar
6.Northcott, D. G., Generalized Koszul complexes and Artinian modules, Quart. J. Math. Oxford Ser. (2) 23 (1972), 289–297.CrossRefGoogle Scholar
7.Sharp, R. Y., Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. (2) 34 (1986), 212–218.CrossRefGoogle Scholar