Lemma 10.40.2. Let $R$ be a ring. Let $M$ be an $R$-module. Then

** A module over a ring has empty support if and only if it is the trivial module. **

\[ M = (0) \Leftrightarrow \text{Supp}(M) = \emptyset . \]

**Proof.**
Actually, Lemma 10.23.1 even shows that $\text{Supp}(M)$ always contains a maximal ideal if $M$ is not zero.
$\square$

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## Comments (1)

Comment #880 by Konrad Voelkel on

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