# Additive identity

In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

## Elementary examples

* The additive identity familiar from elementary mathematics is zero, denoted 0. For example,
    $5+0=5=0+5.$
* In the natural numbers ⁠$\mathbb {N}$⁠ (if 0 is included), the integers ⁠$\mathbb {Z} ,$⁠ the rational numbers ⁠$\mathbb {Q} ,$⁠ the real numbers ⁠$\mathbb {R} ,$⁠ and the complex numbers ⁠$\mathbb {C} ,$⁠ the additive identity is 0. This says that for a number n belonging to any of these sets,
    $n+0=n=0+n.$

## Formal definition

Let N be a group that is closed under the operation of addition, denoted +. An additive identity for N, denoted e, is an element in N such that for any element n in N,
    $e+n=n=n+e.$

## Further examples

* In a group, the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof).
* A ring 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).
* In the ring M$_{m×n}$(R) of m-by-n matrices over a ring R, the additive identity is the zero matrix, 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 ⁠$\operatorname {M} _{2}(\mathbb {Z} )$⁠ the additive identity is
    $0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}$
* In the quaternions, 0 is the additive identity.
* In the ring of functions from ⁠$\mathbb {R} \to \mathbb {R}$⁠, the function mapping every number to 0 is the additive identity.
* In the additive group of vectors in ⁠$\mathbb {R} ^{n},$⁠ the origin or zero vector is the additive identity.

## Properties
### The additive identity is unique in a group

Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G,
    $0+g=g=g+0,\qquad 0'+g=g=g+0'.$
It then follows from the above that
    ${\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.$

### The additive identity annihilates ring elements

In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in S, s · 0 = 0. This follows because:
    ${\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}$

### The additive and multiplicative identities are different in a non-trivial ring

Let R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let r be any element of R. Then
    $r=r\times 1=r\times 0=0$
proving that R is trivial, i.e. R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.