Centrally Essential Endomorphism Rings of Abelian Groups

TL;DR

This paper investigates Abelian groups with centrally essential endomorphism rings, revealing conditions under which these rings are commutative or non-commutative, and providing specific examples of such groups.

Contribution

It characterizes when the endomorphism ring of an Abelian group is commutative or non-commutative and offers new examples of torsion-free groups with non-commutative rings.

Findings

Endomorphism rings are commutative for torsion or non-reduced groups.

Existence of torsion-free groups with non-commutative endomorphism rings.

Provides explicit examples of such groups.

Abstract

We study Abelian groups $A$ with centrally essential endomorphism ring $\text{End}\,A$. If $A$ is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\,A$ is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings. The paper will appear in Communications in Algebra.