---
tag: MDS8655N0Z
kind: statement
role: theorem
name: Goldie's theorem
msc: [16P60, 16S85, 16N60]
uses: [5F0YMCV50R, Z9TJSZZYG1, QFDC3NYHJE, C5971PGKPH, NY25HXA478, 62M0NC3E2T, 20QEAM8AX2, R1F271180N, 0670T59HGE, 8MS51FASMX, TJP4ARC2X7]
about: [QFDC3NYHJE, C5971PGKPH]
sources:
  - url: https://en.wikipedia.org/wiki/Goldie%27s_theorem
same_as:
  wikidata: Q5580171
  wikipedia: Goldie's theorem
---

## Statement

Let $R$ be a ring with identity.

1. $R$ is [[semiprime|5F0YMCV50R]] and right [[Goldie|QFDC3NYHJE]] if and only if $R$ has a [[semisimple|NY25HXA478]] right [[classical ring of quotients|C5971PGKPH]].
2. $R$ is [[prime|Z9TJSZZYG1]] and right Goldie if and only if $R$ has a [[simple|62M0NC3E2T]] [[Artinian|20QEAM8AX2]] right classical ring of quotients.

## Remarks

For the forward direction, one shows that in a semiprime right Goldie ring a right ideal is [[essential|8MS51FASMX]] exactly when it contains a regular element.
From this one gets that the regular elements satisfy the right [[Ore condition|TJP4ARC2X7]], so $Q$ exists.
One also gets that $Q$ has no proper essential right ideals, which forces $Q$ to be semisimple.
For the converse, the Goldie conditions and semiprimeness pass from the semisimple ring $Q$ down to its right order $R$.

Example: a commutative integral domain $R$ is prime and Goldie, and its classical ring of quotients is its field of fractions, a simple Artinian ring.
Example: every semiprime right [[Noetherian ring|0670T59HGE]] has a semisimple right classical ring of quotients, since right Noetherian rings are right Goldie.
So a Weyl algebra over a field, which is a Noetherian domain, has a division ring of quotients.

The semiprime hypothesis is needed.
The ring $\Z/4\Z$ is Noetherian, hence Goldie, but it is not semiprime, because the ideal $2\Z/4\Z$ has square $0$.
Its regular elements are $1$ and $3$, which are already units, so its classical ring of quotients is $\Z/4\Z$ itself, which is not semisimple.

## Explanations

### L1 human

Every integral domain sits inside a field of fractions, and this is how one builds $\Q$ from $\Z$.
Goldie's theorem is the noncommutative version.
A ring with no nilpotent ideals and with mild finiteness conditions sits inside a ring of fractions, and this ring of fractions is as nice as possible.
By the [[Artin–Wedderburn theorem|R1F271180N]], it is a finite product of matrix rings over division rings.

### L2 human

Goldie's theorem is the starting point of the theory of noncommutative Noetherian rings.
It lets one study a prime Noetherian ring $R$ through its quotient ring $Q \cong M_n(D)$, with $D$ a division ring, and define invariants such as the Goldie rank $n$.
It applies to enveloping algebras of finite-dimensional Lie algebras over a field and to Weyl algebras, which are Noetherian domains.