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:

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