The Stacks project

Definition 12.9.2. Let $\mathcal{A}$ be an abelian category.

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

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