Centrally essential endomorphism rings of abelian groups

Abstract

We study Abelian groups A with centrally essential endomorphism ring $\text{End}\hspace{0.17em}A.$ If A is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\hspace{0.17em}A$ is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings.

Keywords:

• Abelian group
• centrally essential ring
• endomorphism ring

MSC2010 database:

• 16R99
• 20K30

Notes

1 It is clear that the ring R with center C is centrally essential if and only if the module R_C is an essential extension of the module C_C.

2 See 3.1 below.

3 See 3.1 below.

4 cf. [15, Proposition 2.2].

5 A ring is said to be invariant if all its one-sided ideals are ideals.