The equivalent condition to an element being invertible in a ring Show that an element $u$ of a ring is invertible with $v=u^{-1}$ if and only if $uvu=u$ and $v$ is the only element satisfying this condition. This is part of an exercise from Jacobson. 

$u(1-vu+v)u=uvu=u$ and by uniqueness $1-vu+v=v$. Similarly, $u(1-uv+v)u=uvu=u$ implies $1-uv+v=v$. The proof completes. 
