Primary decomposition: Compatibility, independence and linear growth

Author:
Yongwei Yao

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1629-1637

MSC (2000):
Primary 13E05; Secondary 13C99, 13H99

Published electronically:
November 15, 2001

MathSciNet review:
1887009

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Abstract | References | Similar Articles | Additional Information

Abstract: For finitely generated modules over a Noetherian ring , we study the following properties about primary decomposition: (1) The Compatibility property, which says that if and is a -primary component of for each , then ; (2) For a given subset , is an open subset of if and only if the intersections for all possible -primary components and of ; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals of there exists a such that for any there exists a primary decomposition of such that every -primary component of that primary decomposition contains .

**[Bo]**Nicolas Bourbaki,*Elements of mathematics. Commutative algebra*, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR**0360549****[Br]**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**[Ei]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[HH]**Melvin Hochster and Craig Huneke,*Tight closure, invariant theory, and the Briançon-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), no. 1, 31–116. MR**1017784**, 10.1090/S0894-0347-1990-1017784-6**[Hu]**Craig Huneke,*Uniform bounds in Noetherian rings*, Invent. Math.**107**(1992), no. 1, 203–223. MR**1135470**, 10.1007/BF01231887**[HRS]**William Heinzer, L. J. Ratliff Jr., and Kishor Shah,*On the embedded primary components of ideals. I*, J. Algebra**167**(1994), no. 3, 724–744. MR**1287067**, 10.1006/jabr.1994.1209**[Mc]**Stephen McAdam,*Primes associated to an ideal*, Contemporary Mathematics, vol. 102, American Mathematical Society, Providence, RI, 1989. MR**1029029****[Ra]**L. J. Ratliff Jr.,*On asymptotic prime divisors*, Pacific J. Math.**111**(1984), no. 2, 395–413. MR**734863****[Sh1]**Rodney Y. Sharp,*Linear growth of primary decompositions of integral closures*, J. Algebra**207**(1998), no. 1, 276–284. MR**1643102**, 10.1006/jabr.1998.7436**[Sh2]**R. Y. Sharp,*Injective modules and linear growth of primary decompositions*, Proc. Amer. Math. Soc.**128**(2000), no. 3, 717–722. MR**1641105**, 10.1090/S0002-9939-99-05170-9**[Sw]**Irena Swanson,*Powers of ideals. Primary decompositions, Artin-Rees lemma and regularity*, Math. Ann.**307**(1997), no. 2, 299–313. MR**1428875**, 10.1007/s002080050035

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Additional Information

**Yongwei Yao**

Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

Email:
yyao@math.ukans.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06284-0

Keywords:
Primary decomposition,
Linear Growth,
Artin-Rees number

Received by editor(s):
October 5, 2000

Received by editor(s) in revised form:
January 12, 2001

Published electronically:
November 15, 2001

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2001
American Mathematical Society