By the definition of "left inverse", ba= I, the identity. By the definition of "right inverse", ac= I.be a system with identity in which is associative. If is a left-inverse for and c a right-inverse for a, then b = c. As corollaries, show that (a) a two inverse is unique, and (b) if is commutative, then has at most one left inverse."
I'm sure there is a super simple solution but it isn't immediately apparent to me, thanks in advance.
Mutiply both sides of ac= I, on the left, by b and see what happens. (Using, of course, the fact that O is associative so b(ac)= (ba)c.)
The last two parts, then, are simple.