Goldie's theorem

Remarks

For the forward direction, one shows that in a semiprime right Goldie ring a right ideal is essential exactly when it contains a regular element. From this one gets that the regular elements satisfy the right Ore condition, 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 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.

Going deeper

For a student

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.