The Stacks project

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. $\square$

Comments (1)

Comment #880 by on

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

There are also:

  • 2 comment(s) on Section 10.40: Supports and annihilators

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View Lemma 10.40.2 as pdf