On lifting modules which do not satisfy the finite internal exchange property

TL;DR

This paper investigates whether all lifting modules satisfy the finite internal exchange property, providing characterizations for certain modules and ultimately demonstrating that the property does not hold universally for lifting modules.

Contribution

It offers new characterizations for when the square of a hollow and uniform module is lifting and resolves the open problem negatively.

Findings

Not all lifting modules satisfy the finite internal exchange property.

Characterizations for the square of hollow and uniform modules being lifting.

The open problem is answered negatively.

Abstract

In this paper, we consider the open problem: does any lifting module satisfy the finite internal exchange property? We give characterizations for the square of a hollow and uniform module to be lifting, and solve the above problem negatively as an application of this result.