This is the exercise problem of John.M. Howie book 5.4
Let $S$ be an semigroup and $J(a)$ is denoted as a principal ideal generated by $a$, where $a \in S.$ Two element $a, b$ are $\mathcal J$ related iff $J(a) = J(b)$ and two classes $J_a \leq J_b$ iff $J(a) \subseteq J(b)$. The minimal $\mathcal J$- class is called kernel of $S$ and it is denoted by $K(S)$. $K(S), J_a \cup \{0\}$ are the principal factors of $S$.
Use Theorem 3.1.6, every principal factor besides kernel is either $0$-simple or null semigroup. $J_a \cup \{0\}$ is not a null semigroup, because $a^{-1} \in J_a$ and $aa^{-1}a = a$. so $J_a \cup \{0\}$ is $0$-simple semigroup. I know that every completely $0$-simple inverse semigroup is isomorphic to Brandt semigroup. It is enough to show that $J_a \cup \{0\}$ contains a primitive idempotent.
How to prove that $J_a \cup \{0\}$ contains primitive idempotents?