Left and Right Zero are the Same

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $z_L \in S$ be a left zero, and $z_R \in S$ be a right zero.

Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a zero $z$.

Furthermore, $z$ is the only left and right zero for $\circ$.

Proof

Let $\struct {S, \circ}$ be an algebraic structure such that:

$\exists z_L \in S: \forall x \in S: z_L \circ x = z_L$

$\exists z_R \in S: \forall x \in S: x \circ z_R = z_R$

Then $z_L = z_L \circ z_R = z_R$ by both the above, hence the result.

The uniqueness of the left and right zero is a direct result of Zero Element is Unique.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $6$