Definition 1

Let M be a module over a ring A. A prime ideal p is said to be associated with M if there exists x ∈ M such that p is equal to the annihilator & x. The set of prime ideals associated with M is denoted by Ass_A(M), or simply Ass(M).

\* Example. Let a be an ideal in the polynomial ring A = \mathbf{C}[X_1, ..., X_n], V the corresponding affine algebraic variety and V_1, ..., V_p the irreducible components of V. If M is taken to be the ring A/a of functions which are regular on V, the set of prime ideals associated with M consists of the ideals of V_1, ..., V_p and in general other prime ideals each of which contains one of the ideals of the V_i. \*

As the annihilator of 0 is A, an element x ∈ M whose annihilator is a prime ideal is necessarily ≠ 0. To say that a prime idea p is associated with M

(*) The results of this chapter depend only on Books I to VI and Chapters I to III of this Book, excluding Chapter I, § 4 and Chapter III, § 5.

amounts to saying that $M$ contains a submodule isomorphic to $A/p$ (namely $Ax$, for all $x \in M$ whose annihilator is $p$).

If an $A$-module $M$ is the union of a family $(M_i)_{i \in I}$ of submodules, then clearly