# Definition:Zero Element

## Definition

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

### Left Zero

An element $z_L \in S$ is called a **left zero element** (or just **left zero**) if and only if:

- $\forall x \in S: z_L \circ x = z_L$

### Right Zero

An element $z_R \in S$ is called a **right zero element** (or just **right zero**) if and only if:

- $\forall x \in S: x \circ z_R = z_R$

### Zero

An element $z \in S$ is called a **two-sided zero element** (or simply **zero element** or **zero**) if and only if it is both a **left zero** and a **right zero**:

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

## Ring and Field Zero

### Ring Zero

Let $\struct {R, +, \circ}$ be a ring.

The identity for ring addition is called the **ring zero** (of $\struct {R, +, \circ}$).

It is denoted $0_R$ (or just $0$ if there is no danger of ambiguity).

### Field Zero

Let $\struct {F, +, \times}$ be a field.

The identity for field addition is called the **field zero** (of $\struct {F, +, \times}$).

It is denoted $0_F$ (or just $0$ if there is no danger of ambiguity).

## 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

- Results about
**zero elements**can be found**here**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.3$. Units and zeros: Definition $2$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids: Exercise $(6)$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**zero element**