Posner's theorem

Remarks

Posner proved that a prime PI ring has a classical ring of quotients that is a simple Artinian ring finite-dimensional over its centre. The sharper statement that one only has to invert the nonzero central elements, given in parts 1 and 2, is a later refinement based on central polynomials. One way to see the original statement: a prime PI ring satisfies the conditions of a Goldie ring, so Goldie's theorem gives a simple Artinian quotient ring $Q \cong M_s(D)$. The ring $Q$ satisfies the same multilinear identities as $R$, and then Kaplansky's theorem shows that $Q$ is finite-dimensional over its centre.

Example: $R = M_n(\Z)$ is prime and satisfies $s_{2n}$, its centre is $\Z$, and $Q = M_n(\Q)$, a central simple algebra over $\Q$.

The primeness hypothesis is needed for the conclusion that $Q$ is simple. The ring $\Z \times \Z$ is commutative, hence PI, but it is not prime, and its classical ring of quotients $\Q \times \Q$ is not simple.

Going deeper

For a student

Posner's theorem reduces the study of prime PI rings to orders in central simple algebras. The dimension of $Q$ over $F$ is a square $n^2$, and $n$ is called the PI degree of $R$. Combined with the theory of generic matrix algebras, it is the main tool for questions about prime PI rings, such as their dimension theory and their centres.