Additive identity

Informally, in mathematics the additive identity of a set of numbers is a number which, on addition to any number n, yields n. In other words, addition of the additive identity to any number n preserves the number.

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.

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 group of m by n matrices over a group G, M_m×n(G) the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in G. For example, in the 2 by 2 matrices over the integers M₂(Z) the additive identity is

0 = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix}

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 and 0' + (0) = (0) = (0) + 0'

which shows that 0 = 0'.

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.

External links

References

David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0471433349.

Abstract algebra | Elementary algebra | Group theory | Ring theory

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