Throughout is a ring with identity element.

Definition 1: is called left (right)artinianif any descending chain of the left (right) ideals of stabilizes. Clearly every finite ring is left and right artinian.

Definition 2: is calledreducedif it has no non-zero nilpotent element, i.e. if for some and integer then

Artin-Wedderburn Theorem: Every reduced left (or right) artinian ring is a finite direct product of matrix rings over some division rings, that is:

for some integers and division rings

Wedderburn's little theorem: Every finite division ring is a field.

Question: Give a short proof of the following problem, which is a special case of Jacobson's difficult theorem:

Let be afinitering and suppose that for any there exists an integer such that Prove that is commutative.