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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 $I_{1}, \ldots ,I_{g}$ be ideals of the commutative ring $R$, let $M$ be a Noetherian $R$-module and let $N$ be a submodule of $M$; also let $A$ be an Artinian $R$-module and let $B$ be a submodule of $A$. It is shown that, whenever $\left (a_{m}\left (1\right ),\ldots ,a_{m}\left (g\right )\right )_{m\in \mathbb {N}}$ is a sequence of $g$-tuples of non-negative integers which is non-decreasing in the sense that $a_{i}\left (j\right )\leq a_{i+1}\left (j\right )$ for all $j=1,\ldots ,g$ and all $i\in \mathbb {N}$, then Ass$_{R}\left (M/ I_{1}^{a_{n}\left (1\right )}\ldots I_{g}^{a_{n}\left (g\right )}N\right )$ is independent of $n$ for all large $n$, and also Att$_{R}\left (B:_{A}I_{1}^{a_{n}\left (1\right )}\ldots I_{g}^{a_{n}\left (g\right )}\right )$ is independent of $n$ for all large $n$. These results are proved without any regularity conditions on the ideals $I_{1}, \ldots ,I_{g}$, and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.


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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

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