---
tag: ZQ9ABBKF1N
kind: concept
name: Progenerator
msc: [16D90, 16D40]
specializes: [BFFA03GE3V]
sources:
  - url: https://en.wikipedia.org/wiki/Morita_equivalence
---

## Summary

A progenerator over a [[ring|0B1261B5ED]] $R$ is a [[finitely generated|05QY1VEBYD]] [[projective module|BFFA03GE3V]] that is also a generator of the module category.
The free modules $R^n$ with $n \ge 1$ are progenerators.
Progenerators are the modules that control [[Morita equivalence|RD5WKA1VK7]].
A ring $S$ is Morita equivalent to $R$ exactly when it is the endomorphism ring of a progenerator over $R$.
This holds up to the choice of side.

## Explanations

### L1 human

The ring $R$ itself, seen as a module, has three key features.
It is generated by one element, it is projective, and every module is a quotient of a sum of copies of it.
A progenerator is any module that shares these features, with finitely many generators allowed.
For example, $R^3$ is a progenerator, since $R$ is a direct summand of it.
The ring of endomorphisms of a progenerator has the same module theory as $R$.