Prove: If [itex]x[/itex] has a right inverse given by [itex]a[/itex] and a left inverse given by [itex]b[/itex], then [itex]a = b[/itex]. 3. The attempt at a solution One thing that bothers me: how can we even talk about a left inverse or a right inverse without establishing that x is in an algebraic structure? I wrote this in my proof but I'm not sure if it's necessary to do so or even correct. Then my last line is kind of questionable. I'm not entirely sure if xa = bx, then a = b. What if I said b*(x*a) = (b*x)*a = e? Would that justify a = b?