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U–essential prime divisors and sequences over an ideal

Published online by Cambridge University Press:  22 January 2016

Daniel Katz  and
Louis J. Ratliff Jr.
Daniel Katz
Affiliation:
Department of Mathematics, University of Kansas Lawrence, Kansas, 66045
Louis J. Ratliff Jr.
Affiliation:
Department of Mathematics, University of California Riverside, California, 92521
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All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.

In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 103 , October 1986 , pp. 39 - 66
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

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