The $\mathcal{J}$- class of a primitive idempotent in a regular semigroup.

Question

I am currently studying some basic facts of regular semigroups and Green's relations and I got stuck on the following exercise problem.

Let $S$ be a regular semigroup with a primitive idempotent $e$. Show, that $J(e)$ is the least ideal of $S$.

I will appreciate any solutions or ideas. Thanks! :)

Answer 1

This is not true. Take the two-element monoid $\{1, 0\}$. Then $1$ is a primitive idempotent but the $\mathcal{J}$-class of $1$ is not even an ideal.