Right Ideal in Monoids

Question

Let $P$ be a left-cancellative monoid, $q \in P$, and define $qP := \{qp \mid p \in P\}$. I conjecture that $qP = P$ implies $q$ is invertible in $P$ (the converse is trivial). Note that $1 \in P = qP$, so there exists $p \in P$ such that $1 = qp$, so $q$ has a right inverse. However, I don't see how to prove that it has a left inverse.

Answer 1

You have $qp=1$. Multiply by $q$ on the right to get $qpq=q$. Now use left cancellation to get $pq=1$.