Let R be a ring with multiplicative identity 1R. Suppose that R is finite. The elemets xy1, xy2,...xyn are all different. So x y_i=1R for some i. A lemma that is not proven is given. If xyi=1R & yjx=1R, then yi=yj I need to show that yjx=1R. Right now I haven't got much. I took the contrapositive of the lemma, but I still get stuck as I'm not sure where I could go from there with the information that I'm given. The book gives a theorem which states Let R be a ring with identity and a, b of R. If a is a unit each of the equations ax=b & ya=b has a unique solution in R. Then it goes on to state that if the ring is not commutative, x may not be equal to y. But yeah, I'm still stuck.