Self-small products of abelian groups

TL;DR

This paper characterizes self-small products of abelian groups by exploring closure properties and provides a description for finitely generated abelian groups, advancing understanding of their structural properties.

Contribution

It introduces a characterization of self-small products of abelian groups using closure properties, specifically describing finitely generated cases.

Findings

Self-small products are characterized via closure properties.

Finitely generated abelian groups' self-small products are explicitly described.

The paper develops a framework for understanding smallness in abelian groups.

Abstract

For abelian groups $A, B$, $A$ is called $B$-small if the covariant functor $Hom(A,-)$ commutes with all direct sums $B^{(\kappa)}$ and $A$ is self-small provided it is $A$-small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.