The additive identity familiar from is zero, denoted 0. For example,

${\displaystyle 5+0=5=0+5.}$

elementary mathematics

In the ${\displaystyle \mathbb {N} }$ (if 0 is included), the integers ${\displaystyle \mathbb {Z} ,}$ the rational numbers ${\displaystyle \mathbb {Q} ,}$ the real numbers ${\displaystyle \mathbb {R} ,}$ and the complex numbers ${\displaystyle \mathbb {C} ,}$ the additive identity is 0. This says that for a number n belonging to any of these sets,

${\displaystyle n+0=n=0+n.}$

natural numbers

In a , the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof).

group

A or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below).

ring

In the ring M_{m × n}(R) of m-by-n over a ring R, the additive identity is the zero matrix,^[1] denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2×2 matrices over the integers ${\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}$ the additive identity is

${\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}$

matrices

In the , 0 is the additive identity.

quaternions

In the ring of from ${\displaystyle \mathbb {R} \to \mathbb {R} }$, the function mapping every number to 0 is the additive identity.

functions

In the of vectors in ${\displaystyle \mathbb {R} ^{n},}$ the origin or zero vector is the additive identity.

additive group

0 (number)

Additive inverse

Identity element

Multiplicative identity

David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003,  0-471-43334-9.

ISBN

at PlanetMath.

Uniqueness of additive identity in a ring