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Associated Prime Ideals in Non-Noetherian Rings

Published online by Cambridge University Press:  20 November 2018

Juana Iroz  and
David E. Rush
Juana Iroz
Affiliation:
University of California, Riverside, California
David E. Rush
Affiliation:
University of California, Riverside, California
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The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 344 - 360
Copyright
Copyright © Canadian Mathematical Society 1984

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