A problem on non-commutative ring

Question

Let $R$ be a non-commutative ring with $1$ and $a,b\in R$ such that $ab=1 \neq ba.$ Could anyone advise me on how to show there exists $c\in R-\{b\}$ such that $ac=1 \ ?$

Hints will suffice. Thank you.

Answer 1

Define a homomorphism of groups $R \rightarrow R$ which is multiplication by $a$ on the left, i.e. $x \mapsto ax$. Then $1$ and $ba$ have the same image.

Then you can conclude...