## A semigroup can have at most one zero

Show that a semigroup can have at most one zero.

Recall that an element $z \in S$ is called a left zero if $zs = z$ for all $s \in S$ and a right zero if $sz = z$ for all $s \in S$, and a zero if it is both a left and a right zero.

Suppose $z$ is a left zero and $w$ a right zero. Then $w = zw = z$. In particular, if $z$ and $w$ are zeros in $S$, then $z = w$.