submodule N of M is called -primary if . A submodule of the R-module R is -primary as a submodule if and only if it is a -primary ideal; thus, when , the above decomposition is precisely a primary decomposition of an ideal. Taking , the above decomposition says the set of associated primes of a finitely generated module M is the same as when (without finite generation, there can be infinitely many associated primes.) Properties of associated primes Let be a Noetherian ring. Then The set of zero-divisors on R is the same as the union of the associated primes of R (this is because the set of zerodivisors of R is the union of the set of annihilators of nonzero elements, the maximal elements of which are associated primes). For the same reason, the union of the associated primes of an R-module M is exactly the set of zero-divisors on M, that is, an element r such that the endomorphism is not injective. Given a subset , M an R-module , there exists a submodule such that and . Let be a multiplicative subset, an -module and the set of all prime ideals of not intersecting . Then is a bijection. Also, . Any prime ideal minimal with respect to containing an ideal J is in These primes are precisely the isolated primes. A module M over R has finite length if and only if M is finitely generated and consists of maximal ideals. Let be a ring homomorphism between Noetherian rings and F a B-module that is flat over A. Then, for each A-module E, . Non-Noetherian case The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals. The proof is given at Chapter 4 of Atiyah–MacDonald as a series of exercises. There is the following uniqueness theorem for an ideal having a primary decomposition. Now, for any commutative ring R, an ideal I and a minimal prime P over I, the pre-image of I RP under the localization map is the smallest P-primary ideal containing I. Thus, in the setting of preceding theorem, the primary ideal Q corresponding to a minimal prime P is also the smallest P-primary ideal containing I and is called the P-primary component of I. For example, if the power Pn of a prime P has a primary decomposition, then its P-primary component is