Throughout is a ring with identity element.
Definition 1: is called left (right) artinian if any descending chain of the left (right) ideals of stabilizes. Clearly every finite ring is left and right artinian.
Definition 2: is called reduced if 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 a finite ring and suppose that for any there exists an integer such that Prove that is commutative.