# Additive identity

In mathematics the **additive identity** of a set which 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.

## Contents

## Elementary examples

- The additive identity familiar from elementary mathematics is zero, denoted 0. For example,
- 5 + 0 = 5 = 0 + 5

- 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 M_{2}(**Z**) 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
**R**^{n}, the origin or zero vector is the additive identity.

## Proofs

### 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:

### 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.

## See also

## References

- David S. Dummit, Richard M. Foote,
*Abstract Algebra*, Wiley (3d ed.): 2003, ISBN 0-471-43334-9 [Amazon-US | Amazon-UK].

## External links

- uniqueness of additive identity in a ring at PlanetMath.org.
- Margherita Barile, "Additive Identity",
*MathWorld*.