Characterization of the inessential endomorphisms in the category of Abelian groups

An endomorphism $f$ of an Abelian group $A$ is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains $A$ as a subgroup. In this paper we show that $f$ is as above if and only if $(f-v\operatorname{id}_A)(A)$ is contained in the maximal divisible subgroup of $A$ for some $v \in \mathbb{Z}$.