## Exhibit a 0-simple semigroup

Give an example of a 0-simple semigroup.

Recall that a semigroup $S$ with zero is called 0-simple if $S^2 \neq 0$ and $S$ has no ideals other than 0 and $S$.

Let $G$ be a group, and consider the semigroup $S = G_0$. (Recall that if $T$ is a semigroup, $T_0$ is the semigroup obtained by attaching a new element $0$ to $T$ which acts like a zero.) So $S$ is a semigroup with zero. Certainly $S^2 \neq 0$. Now let $J \subseteq S$ be an ideal. If $J \neq 0$, then there exists an element $u \in J$ such that $u \neq 0$. Now $Gu = G$ (since $G$ is a group). So $G = Gu \subseteq SJ \subseteq J$, and certianly $0 \in J$. So $J = S$. Thus $S = G_0$ is 0-simple.