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I am quite curious about the definition and applications of the primary decomposition for modules.

  1. The definition of a primary submodule. (Let's assume we work over a commutative noetherian ring $R$ and an $R$-module $M$) When I first worked on Atiyah-Macdonald I used this definition:

A submodule $N$ of $M$ is primary if any zero divisor on $M/N$ is nilpotent.

But recently I saw the definition in Matsumura's commutative algebra, which is slightly different:

A submodule $N$ of $M$ is primary if any zero divisor on $M/N$ is locally nilpotent, i.e. if $a$ is a zero divisor, then for any $x \in M/N$, there exists $n$ possibliy depending on $x$ such that $a^n x = 0$.

Of course, these two definitions agree when $M$ is a finite $R$-module. (which I guess is the most interesting case) But what should be the "right" definition in the general situation?

  1. The application of this. Is this generality of any use? If M is finite, then I know it admits a filtration with quotients being $R/{\mathfrak{p}_i}$ where $\mathfrak{p}_i$ are associated primes. This seems to be quite useful in some proofs. But what about the case where M is infinite?

3 Geometric meaning. Primary decomposition of an ideal I in R is related to the irreducible components of $\mathrm{Spec}(R/I)$. Is there something similar for the module case?

Thanks very much!

Edit: As there still does not seem to be a clear consensus of answers, it would be great if experts could weigh in.

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I removed the newbie --- true newbies won't even think about using such a tag :) – Ilya Nikokoshev Nov 3 '09 at 9:30
Note: the decision to edit this question was based in part on this meta thread:… – David Corwin Aug 23 '12 at 15:55
Personally, I think the notion of "coprimary module" is more important than the notion of "primary submodule." For non-finitely-generated modules over noetherian rings, the right notion of coprimary seems to be that $M$ is coprimary if it has exactly one associated prime. (This gets used in classifying injective modules.) $N \subset M$ is primary if $M/N$ is coprimary. By this definition, a module is coprimary iff all of its finitely generated submodules are; consequently, it agrees with the definition from Matsumura's Commutative Algebra. – Charles Staats Aug 24 '12 at 1:45
As an additional note, the definition Matsumura gives in Commutative Ring Theory (a different book from Commutative Algebra) agrees with the Atiyah-Macdonald definition. That these two definitions are not equivalent, even for a noetherian base ring, can be seen by considering the $\mathbb Z$-module $\bigoplus_n \mathbb Z / p^n$, which is coprimary only by the weaker definition. – Charles Staats Aug 24 '12 at 1:50

The second definition is the correct one (at least in my opinion). It is similar to the correct notion of defining torsion. For instance one does not in general want to define an abelian group A to be p-torsion iff p^nA = 0 as this rules out for instance the Prufer p-group which should certainly be a torsion group but no fixed power of p will kill all of it. In particular, it is the injective envelope of Z/pZ in the category of abelian groups and so has support = {(p)} which means its single associated prime is (p) also. This makes the Prufer p-group p-coprimary with respect to the second definition which is a "sort of extension" of the fact that a finitely generated module over a noetherian ring is coprimary iff it has at most a single associated prime.

This example with the Prufer p-group extends to indecomposable injective modules over noetherian commutative rings with unit - so I guess my justification is that it makes all such guys coprimary with respect to the relevant ideal and it lines up with the right notion of torsion.

The answer to the second question is yes I think... For instance the right notion of support is somewhat subtle for non-finitely generated modules and (although I've never thought of it this way before) being P-coprimary for some prime ideal P does come up.

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My answer to 1. is that Atyiah-MacDonald probably gives the right definition. Matsumura, like most other books I know, only treats primary decomposition for Noetherian rings. In this case things are simpler: for example the set of primes appearing in a minimal primary decomposition is the set of associated primes, something that fails in the general case. See this question for more details.

Atyiah-MacDonald, instead, gives the uniqueness theorems without the Noetherian hypothesis (although they treat the case of modules only in the exercises). For this reason, some definitions are slightly different. When everything is Noetherian of course there is no difference, but without Noetherian assumptions, I would stick with the definitions of Atyiah-MacDonald.

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Dear Andrea, Do the two defintions really give the same notion in the Noetherian case, when the module $M$ is not finitely generated? E.g. if we consider the $\mathbb Z$-module $\mathbb Q_p/\mathbb Z_p$, as in Greg Stevenson's answer, isn't $0$ a primary submodule of this according to Matsumura's definition, but not according to the AM definition? Or am I confused? Regards, Matthew – Emerton Aug 23 '12 at 20:45

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