---
tag: KP1PP7A6JR
kind: concept
name: Quasi-Frobenius ring
msc: [16L60, 16D50]
specializes: [KQKJENEMSN, 20QEAM8AX2]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-Frobenius_ring
same_as:
  wikidata: Q7269430
  wikipedia: Quasi-Frobenius ring
---

## Summary

A quasi-Frobenius ring is a [[ring|0B1261B5ED]] that is Noetherian and [[self-injective|KQKJENEMSN]] on one side; it is then Artinian and self-injective on both sides.
Over such a ring a module is projective if and only if it is injective.
[[Frobenius algebras|8EVP1K2CZH]], group algebras of finite groups over fields and the rings $\Z/n\Z$ are quasi-Frobenius.
The class generalizes Frobenius algebras from algebras over a field to arbitrary rings.

## Explanations

### L1 human

For most rings, projective modules and injective modules are very different kinds of modules.
In a quasi-Frobenius ring they are the same modules.
The rings $\Z/n\Z$ are examples: they are finite, hence Noetherian, and they are injective over themselves.
The ring $\Z$ is not quasi-Frobenius: $\Z$ is projective over itself but not injective.