# A new version of a theorem of Kaplansky

## TL;DR

This paper extends Kaplansky's theorem by proving a $w$-version involving hereditary torsion theories, broadening the understanding of projective modules over commutative rings.

## Contribution

It introduces and proves a new $w$-version of Kaplansky's theorem for projective modules within hereditary torsion theories.

## Key findings

- Established the $w$-version of Kaplansky's theorem
- Extended the class of modules where Kaplansky's decomposition applies
- Provided new insights into projective modules over commutative rings

## Abstract

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the $w$-version of this theorem, where $w$ is a hereditary torsion theory for modules over a commutative ring.