# Thread: Need help with invertibility proof

1. ## Need help with invertibility proof

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.

2. ## Re: Need help with invertibility proof

What does it mean by definition that $xy$ is invertible?

If you have the definition then you can use the fact that the multiplication is an associative operation.

3. ## Re: Need help with invertibility proof

It means that there exists (xy)^-1 so that (xy)^-1 * xy = 1

4. ## Re: Need help with invertibility proof

Originally Posted by rain1
It means that there exists (xy)^-1 so that (xy)^-1 * xy = 1
Yes, $xy$ is invertible is there exists an unique $z \in R$ such that $(xy)z=1$.
Now, use the associtativity.

5. ## Re: Need help with invertibility proof

You mean like this? $(xy)^{-1}\cdot (xy) = x\cdot (y^{-1}\cdot y)\cdot x^{-1} = x\cdot 1x^{-1} = 1 \cdot 1 =1$? Thats all?

6. ## Re: Need help with invertibility proof

Originally Posted by rain1
You mean like this? $(xy)^{-1}\cdot (xy) = x\cdot (y^{-1}\cdot y)\cdot x^{-1} = x\cdot 1x^{-1} = 1 \cdot 1 =1$? Thats all?
In fact, what I mean is the following.
Suppose $xy$ is invertible then there exists an unique $z \in R$ such that $(xy)z=1$. Since $1=(xy)z=x(yz)$ (here I use the associativity), we find that $x$ is invertible with $yz$ as the inverse element.