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

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,

Further examples

Mm × 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(\Z) the additive identity is

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, It then follows from the above that

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:

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 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.