Automorphism-invariant modules satisfy the exchange property

TL;DR

This paper extends the class of modules known to satisfy the exchange property by proving that automorphism-invariant modules, which include quasi-injective modules, also satisfy this property and possess additional algebraic features.

Contribution

It introduces automorphism-invariant modules as a new class satisfying the exchange property and explores their structural properties such as being clean and satisfying cancellation.

Findings

Automorphism-invariant modules satisfy the exchange property.

Automorphism-invariant modules are clean.

Directly-finite automorphism-invariant modules satisfy cancellation.

Abstract

Warfield proved that every injective module has the exchange property. This was generalized by Fuchs who showed that quasi-injective modules satisfy the exchange property. We extend this further and prove that a module invariant under automorphisms of its injective hull satisfies the exchange property. We also show that automorphism-invariant modules are clean and that directly-finite automorphism-invariant modules satisfy the internal cancellation and hence the cancellation property.