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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 xy

I need to show that y

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.

A lemma that is not proven is given. If xy

_{i}=1_{R}& y_{j}x=1_{R}, then y_{i}=y_{j}I need to show that y

_{j}x=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.

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