# Math Help - Algebra, Problems For Fun (14)

1. ## Algebra, Problems For Fun (14)

Throughout $R$ is a ring with identity element.

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

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

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

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

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

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