# Associated prime

In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a submodule of M. The set of associated primes is usually denoted by ${\displaystyle \operatorname {Ass} _{R}(M)\,}$.

In commutative algebra, associated primes are linked to the Lasker-Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with ${\displaystyle \operatorname {Ass} _{R}(R/J)\,}$.Template:Sfn Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.

## Definitions

A nonzero R module N is called a prime module if the annihilator ${\displaystyle \mathrm {Ann} _{R}(N)=\mathrm {Ann} _{R}(N')\,}$ for any nonzero submodule N' of N. For a prime module N, ${\displaystyle \mathrm {Ann} _{R}(N)\,}$ is a prime ideal in R.Template:Sfn

An associated prime of an R module M is an ideal of the form ${\displaystyle \mathrm {Ann} _{R}(N)\,}$ where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent:Template:Sfn if R is commutative, an associated prime P of M is a prime ideal of the form ${\displaystyle \mathrm {Ann} _{R}(m)\,}$ for a nonzero element m of M or equivalently ${\displaystyle R/P}$ is isomorphic to a submodule of M.

In a commutative ring R, minimal elements in ${\displaystyle \operatorname {Ass} _{R}(M)}$ (with respect to the set-theoretic inclusion) are called isolated primes while the rest of the associated primes (i.e., those properly containing associated primes) are called embedded prime.

A module is called coprimary if xm = 0 for some nonzero m ∈ M implies xnM = 0 for some positive integer n. A nonzero finitely generated module M over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime. A submodule N of M is called P-primary if ${\displaystyle M/N}$ is coprimary with P. An ideal I is a P-primary ideal if and only if ${\displaystyle \operatorname {Ass} _{R}(R/I)=\{P\}}$; thus, the notion is a generalization of a primary ideal.

## Properties

Most of these properties and assertions are given in Template:Harv starting on page 86.

The following properties all refer to a commutative Noetherian ring R:

${\displaystyle 0=M_{0}\subset M_{1}\subset \cdots \subset M_{n-1}\subset M_{n}=M\,}$
such that each quotient Mi/Mi−1 is isomorphic to R/Pi for some prime ideals Pi. Moreover every associated prime of M occurs among the set of primes Pi. (In general not all the ideals Pi are associated primes of M.)

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## 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

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