Primary Decomposition and Associated Primes

If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). Equivalently, a belongs to the radical of the annihilator of M/N , denoted by rM (N). Note that λa cannot be both injective and nilpotent. If so, nilpotence gives aM = a(an−1M) ⊆ N , and injectivity gives an−1M ⊆ N . Inductively, M ⊆ N , so M = N , contradicting the assumption that N is proper. Thus if N is a primary submodule of M , then rM (N) is the set of all a ∈ R such that λa is not injective. Since rM (N) is the radical of an ideal, it is an ideal of R, and in fact it is a prime ideal. For if λa and λb are injective, so is λab = λa ◦ λb. (Note that rM (N) is proper because λ1 is injective.) If P = rM (N), we say that N is P -primary.