Definition:Zero Element

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

Zero
An element $z \in S$ is called a two-sided zero element (or simply zero element or zero) 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.

When discussing an algebraic structure $S$ which has a zero element, then this zero is often denoted $z_S$, $n_S$ or $0_S$.

If it is clearly understood what structure is being discussed, then $z$, $n$ or $0$ are usually used.

Also defined as
Beware of the usage of the term zero element to mean the identity element under the operation of addition:
 * $a + 0 = a = 0 + a$

While zero is indeed a zero element under the operation of multiplication:
 * $a \times 0 = 0 = 0 \times a$

it is not, strictly speaking, a zero element of addition.

Also see

 * Definition:Identity Element


 * Zero Element is Unique