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

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 $A$.A surjection $f : A \to B$ of $\mathcal{A}$ is

*essential*if for every proper subobject $A' \subset A$ we have $f(A') \not= B$.

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