Show that if R is ring with identity, $xy$ and $yx$ have inverse and $xy=yx$ then y has an inverse?

Question

Show that if R is ring with identity, $xy$ and $yx$ have inverse and $xy=yx$ then y has an inverse.

I said we have an $a=xy^{-1}=yx^{-1}$ I did some operations on $xy=yx$ but i couldnt prove

Answer 1

We have that $$(yx)(yx)^{-1}=1$$ By associativity, this is $$y(x(yx)^{-1})=1$$ so a right inverse of $y$ is $x(yx)^{-1}$.

You would also want to know that it has a left inverse. Thus we consider $$(xy)^{-1}(xy)=1$$ so $$((xy)^{-1}x)y=1$$ hence $(xy)^{-1}x$ is a left inverse of $y$. As I'm sure you know, if an element has both a left and a right inverse then they are equal, so $y$ has an inverse.

Answer 2

By hypothesis there exists an $a$ and a $b$ such that $axy=yxb=1$.

Therefore $ax=xb=y^{-1}$.