### Support of a module

In algebra, the support of a module M over a commutative ring A is the set of all prime ideals $\mathfrak\left\{p\right\}$ of A such that $M_\mathfrak\left\{p\right\} \ne 0$. It is denoted by $\operatorname\left\{Supp\right\}\left(M\right)$. In particular, $M = 0$ if and only if its support is empty.

• If $0 \to M\text{'} \to M \to M$ \to 0 be an exact sequence of A-modules. Then
$\operatorname\left\{Supp\right\}\left(M\right) = \operatorname\left\{Supp\right\}\left(M\text{'}\right) \cup \operatorname\left\{Supp\right\}\left(M$).
• If $M$ is a sum of submodules $M_\lambda$, then $\operatorname\left\{Supp\right\}\left(M\right) = \cup_\lambda \operatorname\left\{supp\right\}\left(M_\lambda\right).$
• If $M$ is a finitely generated A-module, then $\operatorname\left\{Supp\right\}\left(M\right)$ is the set of all prime ideals containing the annihilator of M. In particular, it is closed.
• If $M, N$ are finitely generated A-modules, then
$\operatorname\left\{Supp\right\}\left(M \otimes_A N\right) = \operatorname\left\{Supp\right\}\left(M\right) \cap \operatorname\left\{Supp\right\}\left(N\right).$
• If $M$ is a finitely generated A-module and I is an ideal of A, then $\operatorname\left\{Supp\right\}\left(M/IM\right)$ is the set of all prime ideals containing $I + \operatorname\left\{Ann\right\}\left(M\right).$ This is $V\left(I\right)\cap \operatorname\left\{Supp\right\}\left(M\right)$.