Associated prime

In mathematics, an associated prime of a module "M" over a commutative ring "R" is a prime ideal of "R" that is the annihilator of some element of "M".

A module is called coprimary if "xm" = 0 for some nonzero "m" ∈ "M" implies "x"^"n""M" = 0 for some positive integer "n". A finitely generated module over a Noetherian ring is coprimary if and only if it has at most one associated prime.

Properties

*Every non-zero module over a Noetherian ring has at least one associated prime, for example, any maximal element of the set of annihilators of elements of "M" is an associated prime.
*If "M" is a finitely generated module over a Noetherian ring then there is a finite ascending sequence of submodules:: $0=M\_0subset\; M\_1subsetcdotssubset\; M\_\{n-1\}subset\; M\_n=M$:such that each quotient "M"_"i"/"M"_"i−1" is isomorphic to "R"/"P"_"i" for some prime ideals "P"_"i". Moreover every associated prime of "M" occurs among the set of primes "P"_"i". (In general not all the ideals "P"_"i" are associated primes of "M".)

Examples

*If "R" is the ring of integers, then non-trivial free abelian groups and non-trivial abelian groups of prime power order are coprimary.
*If "R" is the ring of integers and "M" a finite abelian group, then the associated primes of "M" are exactly the primes dividing the order of "M".
*The group of order 2 is a quotient of the integers "Z" (considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of "Z".

References

*Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1; 978-0-387-94269-8 | id=MathSciNet | id = 1322960 | year=1995 | volume=150