On the ring of inertial endomorphisms of an abelian group

An endomorphisms $\varphi$ of an abelian group $A$ is said inertial if each subgroup $H$ of $A$ has finite index in $H+\varphi (H)$. We study the ring of inertial endomorphisms of an abelian group. Here we obtain a satisfactory description modulo the ideal of finitary endomorphisms. Also the corresponding problem for vector spaces is considered. For the characterization of inertial endomorphisms of an abelian group see arXiv:1310.4625 . The group of invertible inertial endomorphisms has been studied in arXiv:1403.4193 .