---
tag: 89VBQH8HNS
kind: concept
name: Hereditary ring
msc: ["16E60", "16E10"]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Hereditary_ring
same_as:
  wikidata: Q847887
  wikipedia: Hereditary ring
---

## Summary

A left hereditary ring is a [[ring|0B1261B5ED]] in which every left ideal is a [[projective module|BFFA03GE3V]].
Equivalently, every submodule of a projective left module is projective, so the left [[global dimension|GWA0T1WKCN]] is at most $1$.
Principal ideal domains, Dedekind domains and path algebras over a field of finite quivers without oriented cycles are hereditary.
These rings are the next simplest case after semisimple rings, where every module is projective.

## Explanations

### L1 human

Projective modules are the well-behaved modules, the ones that look like pieces of free modules.
In a hereditary ring this good behaviour is inherited by submodules: a submodule of a projective module is again projective.
The ring of integers $\Z$ is the standard example.
Every subgroup of a free abelian group is free, so subgroups of projective $\Z$-modules are projective.
The ring $\Z/4\Z$ is not hereditary, because its ideal $2\Z/4\Z$ is not projective.