Support of a module

Definition
Let $$A$$ be a commutative unital ring and $$M$$ be a module over $$A$$. The support of $$M$$ is the subset of $$Spec(A)$$ (the spectrum of $$A$$) comprising those prime ideals $$P$$ such that $$M_P \ne 0$$

Here, $$M_P$$ denotes the localization of $$M$$ at the prime ideal $$P$$.

Facts

 * If a prime ideal $$P$$ is contained in the support of $$M$$, then any prime ideal containing $$P$$ is in the support of $$M$$.
 * The support of a module is a union of closed subsets. (This follows from the preceding). Conversely any union of closed subsets, arises as the support of a module.
 * For a finitely generated module, the support of the module equals the Galois correspondent closed set to the annihilator of the module (the ideal that annihilates all elements).