# 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.

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*,

Let (*G*, +) be a group and let 0 and 0' in *G* both denote additive identities, so for any *g* in *G*,

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:

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