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

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 𝐴𝑠𝑠(𝑀/𝐼ⁿ𝑀)*, Proc. Amer. Math. Soc.**74**(1979), no. 1, 16–18. MR**521865**, 10.1090/S0002-9939-1979-0521865-8**[K-M-R]**Daniel Katz, Stephen McAdam, and L. J. Ratliff Jr.,*Prime divisors and divisorial ideals*, J. Pure Appl. Algebra**59**(1989), no. 2, 179–186. MR**1007920**, 10.1016/0022-4049(89)90133-3**[Ma]**Eben Matlis,*Injective modules over Noetherian rings*, Pacific J. Math.**8**(1958), 511–528. MR**0099360****[Mc]**Stephen McAdam,*Primes associated to an ideal*, Contemporary Mathematics, vol. 102, American Mathematical Society, Providence, RI, 1989. MR**1029029****[N]**D. G. Northcott,*An introduction to homological algebra*, Cambridge University Press, New York, 1960. MR**0118752****[Ra]**L. J. Ratliff Jr.,*On prime divisors of 𝐼ⁿ, 𝑛 large*, Michigan Math. J.**23**(1976), no. 4, 337–352 (1977). MR**0457421****[Ru]**David E. Rush,*Asymptotic primes and integral closure in modules*, Quart. J. Math. Oxford Ser. (2)**43**(1992), no. 172, 477–499. MR**1188387**, 10.1093/qmathj/43.4.477**[S1]**R. Y. Sharp,*Asymptotic behaviour of certain sets of attached prime ideals*, J. London Math. Soc. (2)**34**(1986), no. 2, 212–218. MR**856506**, 10.1112/jlms/s2-34.2.212**[S2]**R. Y. Sharp,*A method for the study of Artinian modules, with an application to asymptotic behavior*, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 443–465. MR**1015534**, 10.1007/978-1-4612-3660-3_25**[S3]**R. Y. Sharp,*Artinian modules over commutative rings*, Math. Proc. Cambridge Philos. Soc.**111**(1992), no. 1, 25–33. MR**1131475**, 10.1017/S0305004100075125**[S-T]**R. Y. Sharp and Y. Tiraş,*Asymptotic behaviour of integral closures of ideals relative to Artinian modules*, J. Algebra**153**(1992), no. 1, 262–269. MR**1195413**, 10.1016/0021-8693(92)90155-F

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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:
http://dx.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