1.6 Support Theory
The annihilator of a module M is the ideal of elements \(r \in R\) such that \(rm = 0\) for all \(m \in M\).
The support of a module is the set of prime ideals \(\mathfrak {p} \subset R\) such that \(M_{\mathfrak {p}} \neq 0\).
The zero locus of the annihilator of a module is the same as the support of that module. I.e. the set of prime ideals containing the \(\operatorname {Ann}_R(M)\) is the support of \(M\).
Proof
omitted for now