Let $R$ be a ring with unity $1$ and $r \in R$ Then $r$ is left invertible if there exists $q \in R$ with $qr=1$ $q$ is a left inverse of $r$ Similarly, $r$ is right invertible if there exists $s \in R$ with $rs=1$ $s$ is a right inverse of $r$

Note that, if $r$ is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if $r$ is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two inverses are equal, and $r$ is a unit.