# 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 $n$ possibly 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?

2. 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.

• Note: the decision to edit this question was based in part on this meta thread: tea.mathoverflow.net/discussion/1431/… – 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

• 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