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

Question #17100

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

Expert's answer

Consider any chain

*I*1*⊇**I*2*⊇**· · ·*of left ideals in*R*. Then M*n*(*I*1)*⊇*M*n*(*I*2)*⊇**· · ·*is a chain of left ideals in M*n*(*R*), so it must become stationary. This implies that*I*1*⊇**I*2*⊇**· · ·*also becomes stationary, so*R*is left artinian. So,*R*must be a division ring.Need a fast expert's response?

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