## Exhibit the left and right ideals of a zero semigroup and a left zero semigroup

Find all of the left- and right ideals of a zero semigroup and of a left zero semigroup.

Let $S$ be a zero semigroup. That is, there exists $0 \in S$ such that $ab = 0$ for all $a,b \in S$. In particular, $S$ is commutative, so that left and right ideals coincide.

Let $A \subseteq S$ be a subset containing 0. (In particular, $A \neq \emptyset.) Then for all$latex a \in A\$ and $s \in S$, we have $sa = 0 \in A$. So $SA \subseteq A$, and thus $A$ is an ideal.

Now let $S$ be a left zero semigroup. That is, for all $a,b \in S$, we have $ab = a$.

Suppose $A$ is a left ideal. Since $A \neq \emptyset$, there exists an element $a \in A$. Then for all $s \in S$, we have $s = sa \in A$. So in fact $A = S$, and $S$ is the only left ideal. (I.e., S is left simple.)

Now let $B \subseteq S$ be a nonempty subset with $b \in B$. For all $s \in S$, we have $bs = b \in B$. So $BS \subseteq B$, and thus $B$ is a right ideal. That is, every nonempty subset of $S$ is a right ideal.