The Stacks project

Definition 12.9.3. Let $\mathcal{A}$ be an abelian category.

  1. We say an object $A$ of $\mathcal{A}$ is Noetherian if and only if it satisfies the ascending chain condition for subobjects.

  2. We say $\mathcal{A}$ is Noetherian if every object of $\mathcal{A}$ is Noetherian.

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