Characterization of the inessential endomorphisms in the category of Abelian groups. (Q1884087)

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 it is shown that \(f\) is inessential if and only if \((f-v\cdot\text{id}_A)(A)\) is contained in the maximal divisible subgroup of \(A\) for some \(v\in\mathbb{Z}\).