Goldie’s theorem

Let $R$ be a ring with an identity. Then $R$ has a right classical ring of quotients $Q$ which is semisimple Artinian if and only if $R$ is a semiprime right Goldie ring. If this is the case, then the composition length of $Q$ is equal to the uniform dimension of $R$.

An immediate corollary of this is that a semiprime right Noetherian ring always has a right classical ring of quotients.

This result was discovered by Alfred in the late 1950’s.