Associated prime

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, an associated prime of a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> M over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R is a type of <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> of R that arises as an <a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">annihilator</a> of a (prime) submodule of M. The set of associated primes is usually denoted by 
 
 
 
 
 Ass
 
 R
 
 
 ⁡
 (
 M
 )
 ,
 
 
 {\displaystyle \operatorname {Ass} _{R}(M),}
 
 and sometimes called the assassin or assassinator of M (word play between the notation and the fact that an associated prime is an annihilator).
In <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, associated primes are linked to the <a href="/facts/Lasker%25E2%2580%2593Noether_theorem/dgPs3BYw">Lasker–Noether primary decomposition</a> of ideals in <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a> <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian rings</a>. Specifically, if an ideal J is decomposed as a finite intersection of <a href="/facts/Primary_ideal/kvY9HXVL">primary ideals</a>, the <a href="/facts/Radical_of_an_ideal/GYm3Pm9C">radicals</a> of these primary ideals are <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a>, and this set of prime ideals coincides with 
 
 
 
 
 Ass
 
 R
 
 
 ⁡
 (
 R
 
 /
 
 J
 )
 .
 
 
 {\displaystyle \operatorname {Ass} _{R}(R/J).}
 
 Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.

Associated prime open-in-new

Associated prime