Question

Is left artinian domain must be a division ring?

Answer 1

For any nonzero a we consider descending chain $aR \supset a^2R \supset \ldots$. It must stop since R is artinian. So $a^nR = a^{n+1}R$ for some n. Then $a^n = a^{n+1}r$ for some r in R. $a^n(1 - ar) = 0$. As a is nonzero, it has to be $ar = 1$. Analogously element a have left inverse. So it is division ring.