Answer to Question #17100 in Abstract Algebra for Hym@n B@ss

Question

Let R be a domain. Show that if Mn(R) is semisimple, then R is a division ring.

Answer 1

Consider any chain I1 ⊇ I2 ⊇ · · · of left ideals in R. Then Mn(I1) ⊇Mn(I2) ⊇ · · · is a chain of left ideals in Mn(R), so it must become stationary. This implies that I1 ⊇ I2 ⊇ · · · also becomes stationary, so R is left artinian. So, R must be a division ring.