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On asymptotic values of certain sets of attached prime ideals

  • A.-J. Taherizadeh (a1)

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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.

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References

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1.Brodmann, M., Asymptotic stability of Ass(M/InM), Proc. Amer. Math. Soc. 74 (1979), 1618.
2.Kirby, D., Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford Ser. (2) 24 (1973), 4757.
3.Kirby, D., Coprimary decomposition of Artinian modules, J. London Math. Soc. (2) 6 (1973), 571576.
4.Macdonald, I. G., Secondary representation of modules over a commutative ring, Symposia Mathematica 11 (1973), 2343.
5.McAdam, S., Asymptotic prime divisors, Lecture Notes in Mathematics 1023 (Springer, 1983).
6.Northcott, D. G., Generalized Koszul complexes and Artinian modules, Quart. J. Math. Oxford Ser. (2) 23 (1972), 289297.
7.Sharp, R. Y., Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. (2) 34 (1986), 212218.

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