## A characterization of principal ideals in a semigroup

Let $S$ be a semigroup and let $a \in S$. Let $L(a)$, $R(a)$, and $J(a)$ be the left, right, and two-sided ideals generated by $a$, respectively. Prove that $L(a) = a \cup Sa$, $R(a) = a \cup aS$, and $J(a) = a \cup Sa \cup aS \cup SaS$.

Recall that by definition, $L(a)$ is the intersection over the class of left ideals which contain $a$. Suppose $L$ is a left ideal containing $a$. Then for all $s \in S$, $sa \in L$, and thus $a \cup Sa \subseteq L$. So $a \cup Sa \subseteq L(a)$. On the other hand, we have $S(a \cup Sa) = Sa \cup SSa \subseteq Sa$ $\subseteq a \cup Sa$. So $a \cup Sa$ is a left ideal containing $a$, and thus $L(a) \subseteq a \cup Sa$. So $L(a) = a \cup Sa$.

The proofs for $R(a)$ and $J(a)$ are similar; show that every (right,two sided) ideal contains the set in question, and that the set in question is a (right, two-sided) ideal.