Let R and B be as in Exercise 12. Then the associated primes of B are precisely the primes P₁, .. . , P_n, where O = A₁ ∩ · · · ∩ A_n is a reduced primary decomposition of O with each A_i P_i-primary. In particular, the set of associated primes of B is finite.

Data from exercise 12

Let R be Noetherian and let B be an R-module satisfying the ascending chain condition on submodules. Then the following are equivalent:

(i) There exists exactly one associated prime of B;

(ii) B ≠ O and for each r ϵ R one of the following is true: either rx = 0 implies x = 0 for all x ϵ B or for each x ϵ B there exists a positive integer n(x) such that r^n(x) = 0.