On the ring of inertial endomorphisms of an abelian group

TL;DR

This paper investigates the structure of inertial endomorphisms in abelian groups, providing a detailed description modulo finitary endomorphisms and extending the analysis to vector spaces.

Contribution

It offers a comprehensive characterization of the ring of inertial endomorphisms of abelian groups, building on previous work and considering the vector space case.

Findings

Description of inertial endomorphisms modulo finitary endomorphisms

Characterization of the ring structure of inertial endomorphisms

Analysis of invertible inertial endomorphisms

Abstract

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 .