Prove that in a finite monoid each element is invertible

Question

Let $(M,\circ)$ be a finite monoid. Suppose the identity element $e\in M$ is the only idempotent element. Then prove that each element in $M$ has inverse.
How can I prove this?

Answer 1

Let $a$ be an element of a finite monoid.

Then $n>0$ and $k>0$ exist with $a^{n+k}=a^{n}$, leading to $a^{m+kr}=a^{m}$ for $m\geq n$ and $r\geq0$.

Choose some $r$ such that $kr\geq n$ and note that $a^{kr}$ is idempotent.

If identity $e$ is unique as idempotent then $a^{kr}=e$, showing that $a^{kr-1}$ serves as inverse of $a$.