Question

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

Answer 1

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