Throughout $\displaystyle R$ is a ring with identity element.

__Definition 1__: $\displaystyle R$ is called left (right) *artinian* if any descending chain of the left (right) ideals of $\displaystyle R$ stabilizes. Clearly every finite ring is left and right artinian.

__Definition 2__: $\displaystyle R$ is called *reduced* if it has no non-zero nilpotent element, i.e. if $\displaystyle x^n=0,$ for some $\displaystyle x \in R$ and integer $\displaystyle n \geq 2,$ then $\displaystyle x=0.$

__Artin-Wedderburn Theorem__: Every reduced left (or right) artinian ring $\displaystyle R$ is a finite direct product of matrix rings over some division rings, that is:

$\displaystyle R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k),$

for some integers $\displaystyle n_j \geq 1$ and division rings $\displaystyle D_j.$

__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 $\displaystyle R$ be a __finite__ ring and suppose that for any $\displaystyle x \in R,$ there exists an integer $\displaystyle n(x) \geq 2$ such that $\displaystyle x^{n(x)}=x.$ Prove that $\displaystyle R$ is commutative.