Additive identity

Elementary examples

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

In the natural numbers N and all of its supersets (the integers Z the rational numbers Q, the real numbers R, or the complex numbers C), the additive identity is 0. Thus for any one of these numbers n, n + 0 = n = 0 + n

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 M_m×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 = ( 0 0 0 0 )

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.

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:

s ⋅ 0 = s ⋅ ( 0 + 0 ) = s ⋅ 0 + s ⋅ 0 ⇒ s ⋅ 0 = s ⋅ 0 − s ⋅ 0 ⇒ s ⋅ 0 = 0

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.