A monoid can have only one zero

Question

An element $z$ of a monoid $M$ is call a zero element if $xz=zx=z$ for all $x\in M$, can a monoid have more than one zero?

I have put my attempt to prove this in the answer below.

Answer 1

A monoid can have only one zero.

Assume that it can have more than one zero.

$xz_1=z_1x=z_1$ and $xz_2=z_2x=z_2$

well then $z_1=z_1z_2=z_2$ hence these are the same, and thus there is only one zero.