The bijection between projective indecomposable and simple modules

TL;DR

This paper provides a clear, self-contained exposition of the canonical bijection between projective indecomposable modules and simple modules over finite-dimensional algebras, elucidating fundamental module theory concepts.

Contribution

It offers a straightforward, self-contained proof of the bijection, making the core correspondence accessible without prior complex background.

Findings

Establishes a canonical bijection between projective indecomposable and simple modules.

Provides a self-contained, accessible proof of the correspondence.

Clarifies foundational concepts in module theory for finite-dimensional algebras.

Abstract

For modules over a finite-dimensional algebra, there is a canonical one-to-one correspondence between the projective indecomposable modules and the simple modules. In this purely expository note, we take a straight-line path from the definitions to this correspondence. The proof is self-contained.