One-sided inverses of finite-dimensional endomorphisms

For endomorphisms $S,T$ of the same finite-dimensional vector space, $ST=I$ makes $S$ surjective and $T$ injective. The rank-nullity theorem makes both invertible, hence $TS=I$. The finite-dimensional and equal-domain assumptions matter; unilateral shift operators on an infinite-dimensional sequence space give one-sided inverses that are not two-sided.