# Answer to Question #16530 in Abstract Algebra for Hym@n B@ss

Question #16530

Prove that a nonzero ring R is a division ring iff every a ∈ R\{0} is right-invertible

Expert's answer

For the “if” part, it suffices to show that

*ab*= 1*⇒**ba*= 1 in*R*. From*ab*=1, we have*b*= 0, so*bc*= 1 for some*c**∈**R*. Now left multiplication by*a*shows*c*=*a*, so indeed*ba*= 1
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