# Answer to Question #17115 in Abstract Algebra for sanches

Question #17115

Let R be any semisimple ring. Show that R is Dedekind-finite, i.e. ab = 1 implies ba = 1 in R.

Expert's answer

By the Wedderburn-Artin Theorem, we are reduced to the case when

Think of

*R*= M*n*(*D*) where*D*is a division ring.Think of

*R*as End(*V*) where_{D}*V*is the space of column*n*-tuples over*D*. If*ab*= 1 in*R*, then clearly ker(*b*) = 0, and this implies that*b**∈*Aut(*V*). In particular,_{D}*ba*= 1*∈**R*.
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