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 (prime) submodule of M. The set of associated primes is usually denoted by ${\displaystyle \operatorname {Ass} _{R}(M),}$ and sometimes called the assassin or assassinator of M (word play between the notation and the fact that an associated prime is an annihilator).^[1]

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 ${\displaystyle \operatorname {Ass} _{R}(R/J).}$^[2] Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.