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 Template:Mvar in the set, yields Template:Mvar. 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,

  $5+0=5=0+5.$

• In the natural numbers Template:Tmath (if 0 is included), the integers Template:Tmath the rational numbers Template:Tmath the real numbers Template:Tmath and the complex numbers Template:Tmath the additive identity is 0. This says that for a number Template:Mvar belonging to any of these sets,

  $n+0=n=0+n.$

Formal definition

Let Template:Mvar be a group that is closed under the operation of addition, denoted +. An additive identity for Template:Mvar, denoted Template:Mvar, is an element in Template:Mvar such that for any element Template:Mvar in Template:Mvar,

$e+n=n=n+e.$

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 ring Template:Math of Template:Mvar-by-Template:Mvar matrices over a ring Template:Mvar, the additive identity is the zero matrix, denoted Template:Math or Template:Math, and is the Template:Mvar-by-Template:Mvar matrix whose entries consist entirely of the identity element 0 in Template:Mvar. For example, in the 2×2 matrices over the integers Template:Tmath the additive identity is

  $0=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

• In the quaternions, 0 is the additive identity.
• In the ring of functions from Template:Tmath, the function mapping every number to 0 is the additive identity.
• In the additive group of vectors in Template:Tmath the origin or zero vector is the additive identity.

Properties

The additive identity is unique in a group

Let Template:Math be a group and let Template:Math and Template:Math in Template:Mvar both denote additive identities, so for any Template:Mvar in Template:Mvar,

$0+g=g=g+0,0^{\prime }+g=g=g+0^{\prime }.$

It then follows from the above that

$0^{\prime }=0^{\prime }+0=0^{\prime }+0=0.$

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 Template:Mvar in Template:Mvar, Template:Math. This follows because:

$\begin{array}{cc}s\cdot 0& =s\cdot (0+0)=s\cdot 0+s\cdot 0\\ \Rightarrow s\cdot 0& =s\cdot 0-s\cdot 0\\ \Rightarrow s\cdot 0& =0.\end{array}$

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

Let Template:Mvar be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let Template:Mvar be any element of Template:Mvar. Then

$r=r\times 1=r\times 0=0$

proving that Template:Mvar is trivial, i.e. Template:Math The contrapositive, that if Template:Mvar 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, Template:ISBN.

External links

• Template:PlanetMath