## A characterization of left simple and simple semigroups

Let $S$ be a semigroup. Show that $S$ is left simple if and only if $Sa = S$ for all $a \in S$, and is simple if and only if $SaS = S$ for all $a \in S$.

Recall that a semigroup is called left simple if the only left ideal is $S$ itself.

Suppose $S$ is left simple. Now $S(Sa) \subseteq Sa$, so that $Sa$ is a left ideal. Hence $Sa = S$. Conversely, suppose $Sa = S$ for every $a \in S$. Let $L$ be a left ideal. Now $L \neq \emptyset$, so we have $a \in L$ for some $a$. Then $S = Sa \subseteq L$, and thus $L = S$. So $S$ is left simple.

Similarly, we can show that $S$ is right simple if and only if $aS = S$ for all $a \in S$.

Now suppose $S$ is simple. (That is, it has no nontrivial ideals.) Let $a \in S$. Now $S(SaS) \subseteq SaS$ and $(SaS)S \subseteq SaS$, so that $SaS$ is an ideal. Hence $SaS = S$. Now suppose $SaS = S$ for all $a \in S$. If $I$ is a two-sided ideal, then we have $a \in I$ for some $a \in S$. Now $S = SaS \subseteq I$, so that $I = S$. Hence $S$ is simple.