Self-pure-generators over Dedekind domains

TL;DR

This paper characterizes when all pure submodules of finite rank torsion-free modules over Dedekind domains are generated by the entire module, linking this property to the existence of a specific rank 1 summand with a particular type.

Contribution

It provides a necessary and sufficient condition involving the inner type for pure submodules to be generated by the whole module in Dedekind domains.

Findings

All pure submodules are $A$-generated iff a rank 1 summand with inner type exists.

The characterization connects module decomposition with type theory.

The result applies specifically to finite rank torsion-free modules over Dedekind domains.

Abstract

We prove that all pure submodules of a finite rank torsion-free module $A$ over a Dedekind domain are $A$-generated if and only if $A$ has a rank $1$ direct summand $B$ such that $\mathbf{type}(B)$ is the inner type of $A$.