Answer to Question #17115 in Abstract Algebra for sanches

Question

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

Answer 1

By the Wedderburn-Artin Theorem, we are reduced to the case when R = Mn(D) where D is a division ring.
Think of R as End(V_D) where 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_D). In particular, ba = 1 ∈ R.