Hello.
I would use some help with following task:
Prove that when in ring R products xy and yx are invertible then elements x and y are also invertible.
Thanks in advance.
In fact, what I mean is the following.
Suppose $\displaystyle xy$ is invertible then there exists an unique $\displaystyle z \in R$ such that $\displaystyle (xy)z=1$. Since $\displaystyle 1=(xy)z=x(yz)$ (here I use the associativity), we find that $\displaystyle x$ is invertible with $\displaystyle yz$ as the inverse element.