Artin’s Theorem

Abstract

Artin’s Theorem says that if a ring R satisfies the polynomial identities satisfied by ${M_n}\left( {\Bbb Z} \right)$ and no nonzero homomorphic image satisfies the polynomial identities satisfied by ${M_{n - 1}}({\Bbb Z})$ then R is Azumaya. Thus Kaplansky’s Theorem, Posner’s Theorem and Artin’s Theorem have a common theme: If R satisfies a polynomial identity (plus a further hypothesis), then R has a large center (plus a further conclusion).