# $\Sigma$-pure-injective modules for string algebras and linear relations

## TL;DR

This paper characterizes indecomposable $ ext{Sigma}$-pure-injective modules over string algebras as either string or band modules, using a splitting result for infinite-dimensional linear relations.

## Contribution

It provides a classification of indecomposable $ ext{Sigma}$-pure-injective modules for string algebras, linking module theory with linear relations.

## Key findings

- Indecomposable $ ext{Sigma}$-pure-injective modules are string or band modules.
- A splitting theorem for infinite-dimensional linear relations is established.
- The result advances understanding of module categories over string algebras.

## Abstract

We prove that indecomposable $\Sigma$-pure-injective modules for a string algebra are string or band modules. The key step in our proof is a splitting result for infinite-dimensional linear relations.