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.