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