## Characterization of the inessential endomorphisms in the category of Abelian groups.(English)Zbl 1061.20047

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 $$v\in\mathbb{Z}$$.

### MSC:

 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups 20K40 Homological and categorical methods for abelian groups 20K27 Subgroups of abelian groups
Full Text: