Asymptotic behaviour

of certain sets of prime ideals

Authors:
Alan K. Kingsbury and Rodney Y. Sharp

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1703-1711

MSC (1991):
Primary 13E05, 13E10

DOI:
https://doi.org/10.1090/S0002-9939-96-03400-4

MathSciNet review:
1328355

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

**[B]**M. Brodmann,*Asymptotic stability of*, Proc. Amer. Math. Soc.**74**(1979), 16--18. MR**80c:13012****[K-M-R]**D. Katz, S. McAdam and L. J. Ratliff, Jr.,*Prime divisors and divisorial ideals*, J. Pure Appl. Algebra**59**(1989), 179--186. MR**90g:13030****[Ma]**E. Matlis,*Injective modules over Noetherian rings*, Pacific J. Math.**8**(1958), 511--528. MR**20:5800****[Mc]**S. McAdam,*Primes associated to an ideal*, Contemporary Mathematics, vol. 102, Amer. Math. Soc., Providence, 1989. MR**90m:13004****[N]**D. G. Northcott,*An introduction to homological algebra*, Cambridge Univ. Press, Cambridge, 1960. MR**22:9523****[Ra]**L. J. Ratliff, Jr.,*On prime divisors of , large*, Michigan Math. J.**23**(1976), 337--352. MR**56:15626****[Ru]**D. E. Rush,*Asymptotic primes and integral closure in modules*, Quart. J. Math. Oxford Ser. (2)**43**(1992), 477--499. MR**93m:13009****[S1]**R. Y. Sharp,*Asymptotic behaviour of certain sets of attached prime ideals*, J. London Math. Soc. (2)**34**(1986), 212--218. MR**87i:13007****[S2]**------,*A method for the study of Artinian modules, with an application to asymptotic behavior*, Commutative algebra (Proceedings of a Microprogram held June 15-July 2, 1987), Mathematical Sciences Research Institute Publications, vol. 15, Springer-Verlag, New York, 1989, pp. 443--465. MR**91a:13011****[S3]**------,*Artinian modules over commutative rings*, Math. Proc. Cambridge Phil. Soc.**111**(1992), 25--33. MR**93a:13009****[S-T]**R. Y. Sharp and Y. Tira\c{s},*Asymptotic behaviour of integral closures of ideals relative to Artinian modules*, J. Algebra**153**(1992), 262--269. MR**94b:13007**

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Additional Information

**Alan K. Kingsbury**

Affiliation:
Pure Mathematics Section, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield, S3 7RH, United Kingdom

Email:
a.kingsbury@sheffield.ac.uk

**Rodney Y. Sharp**

Affiliation:
Pure Mathematics Section, School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield, S3 7RH, United Kingdom

Email:
r.y.sharp@sheffield.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-96-03400-4

Keywords:
Associated prime ideal,
attached prime ideal,
Noetherian module,
Artinian module,
asymptotic prime divisors

Received by editor(s):
December 13, 1994

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1996
American Mathematical Society