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 additive identity is 0. Thus for any one of these numbers n,
• :

  Formal definition

Further examples

• In a group, the additive identity is the identity element of the group, is often denoted 0, and is unique.
• 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 has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial.
• In the ring M_m×n 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 the additive identity is
• :
• 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.

  Properties

The additive identity is unique in a group

The additive identity annihilates ring elements

The additive and multiplicative identities are different in a non-trivial ring