---
tag: 7NA22G672W
kind: concept
name: Semihereditary ring
msc: ["16E60"]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Hereditary_ring
---

## Summary

A left semihereditary ring is a [[ring|0B1261B5ED]] in which every finitely generated left ideal is a [[projective module|BFFA03GE3V]].
Every [[hereditary ring|89VBQH8HNS]] is semihereditary, and the two notions agree for left Noetherian rings.
Commutative semihereditary domains are exactly the [[Prüfer domains|2RDAA4PCK8]], and every [[von Neumann regular ring|DXX7GNY3HE]] is semihereditary.

## Explanations

### L1 human

Semihereditary rings ask for less than hereditary rings: only the finitely generated ideals need to be projective.
This weaker condition matters for rings that are not Noetherian, where some ideals need infinitely many generators.
The ring of all algebraic integers is a good example.
Every finitely generated ideal in it is principal, hence projective, so it is semihereditary.
It is not hereditary, because it has an ideal that is not finitely generated and not projective.