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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 $R$ and an R-module M) $R$-module $M$) When I first worked on Atiyah-Macdonald I used this definition:

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

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

A submodule N $N$ of M $M$ is primary if any zero divisor on M/N $M/N$ is locally nilpotent, i.e. if a $a$ is a zero divisor, then for any $x \in M/NM/N$, there exists n $n$ possibliy depending on x $x$ such that $a^n x = 00$.

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

2

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/p_i $R/{\mathfrak{p}_i}$ where p_i $\mathfrak{p}_i$ 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 Spec R/I. $\mathrm{Spec}(R/I)$. 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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# Primary decomposition for modules

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?

2 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/p_i where 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 Spec R/I. Is there something similar for the module case?

Thanks very much!