Definition:Zero Element

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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

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