In abstract algebra, an associated prime of a moduleM over a ringR 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 .
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 . 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 for any nonzero submodule N' of N. For a prime module N, is a prime ideal in R.
An associated prime of an R module M is an ideal of the form where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent: if R is commutative, an associated prime P of M is a prime ideal of the form for a nonzero element m of M or equivalently is isomorphic to a submodule of M.
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