Invertibility of the product of two elements in a ring

Question

Let $(R,+,·)$ be a non-commutative ring. If $a\in R$ is invertible, and $x\in R$ is non-invertible, does that imply that $ax$ is non-invertible?

Answer 1

If $ax$ was invertible then lets call the inverse $y$. $$yax = axy = e $$

Now lets multiply by $a^{-1}$ $$a^{-1}yax = xy =a^{-1}$$

Now multiply by $a$

$$xya = e$$

Therefore $x^{-1} = ya$

Answer 2

Yes, if $b = (ax)^{-1}$, then $x^{-1} = b a$.