Associated prime

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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 xⁿM = 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.

[edit] 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

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.)

[edit] 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.

[edit] References

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8