Zero Element is Unique

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure that has a zero element $z \in S$.

Then $z$ is unique.

When discussing an algebraic structure $S$ which has a zero element, then this zero is usually denoted $z_S$, $n_S$ or $0_S$.

If it is clearly understood what structure is being discussed, then $z$, $n$ or $0$ are usually used.

Proof
Suppose $z_1$ and $z_2$ are both zeroes of $\left({S, \circ}\right)$.

Then by the definition of zero element:
 * $z_2 \circ z_1 = z_1$ by dint of $z_1$ being a zero;
 * $z_2 \circ z_1 = z_2$ by dint of $z_2$ being a zero.

So $z_1 = z_2 \circ z_1 = z_2$.

So $z_1 = z_2$ and there is only one zero after all.