Zero Element is Unique

Theorem

Let $\struct {S, \circ}$ be an algebraic structure that has a zero element $z \in S$.

Then $z$ is unique.

Proof

Suppose $z_1$ and $z_2$ are both zeroes of $\struct {S, \circ}$.

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.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids: Exercise $(6)$