Definition:Zero Element

Definition
Let $$\left({S, \circ}\right)$$ be an algebraic structure.

Left Zero
An element $$z_L \in S$$ is called a left zero element (or just left zero) iff:
 * $$\forall x \in S: z_L \circ x = z_L$$

Right Zero
An element $$z_R \in S$$ is called a right zero element (or just right zero) iff:
 * $$\forall x \in S: x \circ z_R = z_R$$

Zero
An element $$z \in S$$ is called a two-sided zero element (or simply zero element or zero) iff it is both a left zero and a right zero:
 * $$\forall x \in S: x \circ z = z = z \circ x$$

A zero element is also sometimes called an annihilator, but this term has a more specific definition in the context of linear algebra.

Also see

 * Ring zero
 * Identity element