Uniqueness of inverses in rings.

Question

I can't get example of ring where inverses aren't unique ? What is condition required for the inverses to be unique .
Kindly help..

Answer 1

For any binary operation that is associative and has an identity element, if a left and a right inverse for an element exist, they are necessarily equal.

Indeed, suppose that $ax=e$, $xb=e$. Then $$ a=ae=axb=eb=b. $$ A particular consequence is that inverses are unique in any context where the operation is associative.