Additive identity

Short description: Value that makes no change when added

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.

Elementary examples

• The additive identity familiar from elementary mathematics is zero, denoted 0. For example,

  [math]\displaystyle{ 5+0 = 5 = 0+5. }[/math]

• In the natural numbers [math]\displaystyle{ \N }[/math] (if 0 is included), the integers [math]\displaystyle{ \Z, }[/math] the rational numbers [math]\displaystyle{ \Q, }[/math] the real numbers [math]\displaystyle{ \R, }[/math] and the complex numbers [math]\displaystyle{ \C, }[/math] the additive identity is 0. This says that for a number n belonging to any of these sets,

  [math]\displaystyle{ n+0 = n = 0+n. }[/math]

Formal definition

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,

[math]\displaystyle{ e+n = n = n+e. }[/math]

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 the zero matrix, denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2×2 matrices over the integers [math]\displaystyle{ \operatorname{M}_2(\Z) }[/math] the additive identity is

  [math]\displaystyle{ 0 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} }[/math]

• In the quaternions, 0 is the additive identity.
• In the ring of functions from [math]\displaystyle{ \R \to \R }[/math], the function mapping every number to 0 is the additive identity.
• In the additive group of vectors in [math]\displaystyle{ \R^n, }[/math] the origin or zero vector is the additive identity.

Properties

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,

[math]\displaystyle{ 0+g = g = g+0, \qquad 0'+g = g = g+0'. }[/math]

It then follows from the above that

[math]\displaystyle{ {\color{green}0'} = {\color{green}0'} + 0 = 0' + {\color{red}0} = {\color{red}0}. }[/math]

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 follows because:

[math]\displaystyle{ \begin{align} s \cdot 0 &= s \cdot (0 + 0) = s \cdot 0 + s \cdot 0 \\ \Rightarrow s \cdot 0 &= s \cdot 0 - s \cdot 0 \\ \Rightarrow s \cdot 0 &= 0. \end{align} }[/math]

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

[math]\displaystyle{ r = r \times 1 = r \times 0 = 0 }[/math]

proving that R is trivial, i.e. R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.

See also

• 0 (number)
• Additive inverse
• Identity element
• Multiplicative identity

Bibliography

• David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, ISBN:0-471-43334-9.

External links

• Uniqueness of additive identity in a ring at PlanetMath.org.