Associated prime

（重定向自Associated prime ideal）

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 $\operatorname{Ass}_R(M)\,$ .

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 $\operatorname{Ass}_R(R/J)\,$ . Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.

单词	Associated prime ideal
释义	Associated prime ideal 英语百科 Associated prime （重定向自Associated prime ideal） 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 $\operatorname{Ass}_R(M)\,$ . 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 $\operatorname{Ass}_R(R/J)\,$ . Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.
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