**Theorem.** (Kaplansky-Amitsur) Let be a left primitive algebra, a faithful simple module and If satisfies a polynomial of degree then

i) where So is a field.

ii)

We proved i) in the previous section. So we just need to prove ii). First two easy remarks.

**Remark 1.** Let be a field. Then for any algebras

**Remark 2.** Let be a commutative ring, a PI -algebra and a commutative -algebra. If satisfies a multi-linear polynomial then will also satisfy

*Proof of ii).* So satisfies some multi-linear polynomial of degree at most Clearly satisfies too because it’s a subring of Let be a maximal subfield of and By Remark 2, also satisfies But, by Azumaya theorem, is left primitive, is a faithful simple left -module and Thus, by part i) of the theorem, for some positive integer Therefore by Remark 1

Hence On the other hand, by Remark 2, satisfies and so Therefore and so Finally we have