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 iff:
 * $$\forall x \in S: z_L \circ x = z_L$$

Right Zero
An element $$z_R \in S$$ is called a 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 or simply zero iff it is both a left zero and a right zero:
 * $$\forall x \in S: x \circ z = z = z \circ x$$

Also see

 * Identity Element


 * Zero of a Ring, which is the same thing but in the specific context of a ring.