Chains of Modules, An Artinian Domain is a Division Ring
An Artinian Domain is a Division Ring
Let R be a left artinian domain.
Given a nonzero x in R, consider the descending chain of principal left ideals spanned by the powers of x.
At some point, xn and xn+1 generate the same ideal.
Thus yxn+1 = xn, and yx = 1.
Since every x is left invertible, R is a division ring.