Show that a ring $R$ with $|R|>1$ is a division ring if it satisfies the following condition :
for each $a\neq 0$ in $R$ there exists a unique element $x$ such that $axa=a$
We need to show that $R^*,*$ is a group.
Take $a,b\in R^*$ where $R^*=R\setminus\{0\}$ then $ab\neq 0$
if $ab=0\implies a(b+x)a=aba+axa=a\implies b+x=x\implies b=0$
How to show that for each $a\in R^* , a $ has an inverse.
Please help.