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:

$\forall x \in S: x \circ z = z = z \circ x$

Also known as

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

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.3$: Definition $2$