In mathematics the additive identity of a set which 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.

## Formal definition

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

e + n = n = n + e

Example: The formula is n + 0 = n = 0 + n.

## 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 Mm×n(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices over the integers M2(Z) the additive identity is
$0 = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}$
• In the quaternions, 0 is the additive identity.
• In the ring of functions from R to R, the function mapping every number to 0 is the additive identity.
• In the additive group of vectors in Rn, the origin or zero vector is the additive identity.

## Proofs

### 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 and 0' + g = g = g + 0'

It follows from the above that

(0') = (0') + 0 = 0' + (0) = (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 can be seen because:

\begin{align} 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{align}

### 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, or 0 = 1. Let r be any element of R. Then

r = r × 1 = r × 0 = 0

proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.