The Stacks project

Definition 47.2.1. Let $\mathcal{A}$ be an abelian category.

1. An injection $A \subset B$ of $\mathcal{A}$ is essential, or we say that $B$ is an essential extension of $A$, if every nonzero subobject $B' \subset B$ has nonzero intersection with $A$.

2. A surjection $f : A \to B$ of $\mathcal{A}$ is essential if for every proper subobject $A' \subset A$ we have $f(A') \not= B$.