Definition:Zero Element

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

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.

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 see

 * Definition:Ring Zero
 * Definition:Identity Element


 * Zero Element is Unique