---
tag: QFDC3NYHJE
kind: concept
name: Goldie ring
msc: [16P60, 16U20]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Goldie%27s_theorem
---

## Summary

A right Goldie ring is a [[ring|0B1261B5ED]] with two finiteness conditions.
It contains no infinite direct sum of nonzero right ideals, and it has the ascending chain condition on right [[annihilators|HB4XDKJTW8]].
Every right [[Noetherian ring|0670T59HGE]] is right Goldie, and so is every commutative integral domain.
These are the conditions in [[Goldie's theorem|MDS8655N0Z]], which describes the semiprime right Goldie rings as the rings with a semisimple Artinian [[classical ring of quotients|C5971PGKPH]].

## Explanations

### L1 human

The Goldie conditions are much weaker than the Noetherian condition, but they are strong enough to make the ring behave like a ring with finitely many "directions".
The first condition says that one cannot place infinitely many independent right ideals side by side.
The second condition limits only the special right ideals that arise as annihilators.
For example, an infinite product of copies of a field is not Goldie, because the copies of the field in each coordinate give infinitely many independent ideals.