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

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

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

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

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