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 {{mvar|x}} in the set, yields {{mvar|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,
: $5+0 = 5 = 0+5.$
In the natural numbers {{tmath|\N}} (if 0 is included), the integers {{tmath|\Z,}} the rational numbers {{tmath|\Q,}} the real numbers {{tmath|\R,}} and the complex numbers {{tmath|\C,}} the additive identity is 0. This says that for a number {{mvar|n}} belonging to any of these sets,
: $n+0 = n = 0+n.$

Formal definition

Let {{mvar|N}} be a group that is closed under the operation of addition, denoted +. An additive identity for {{mvar|N}}, denoted {{mvar|e}}, is an element in {{mvar|N}} such that for any element {{mvar|n}} in {{mvar|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 ring {{math|M_{m × n}(R)}} of {{mvar|m}}-by-{{mvar|n}} matrices over a ring {{mvar|R}}, the additive identity is the zero matrix,{{Cite web|last=Weisstein|first=Eric W.|title=Additive Identity|url=https://mathworld.wolfram.com/AdditiveIdentity.html|access-date=2020-09-07|website=mathworld.wolfram.com|language=en}} denoted {{math|O}} or {{math|0}}, and is the {{mvar|m}}-by-{{mvar|n}} matrix whose entries consist entirely of the identity element 0 in {{mvar|R}}. For example, in the 2×2 matrices over the integers {{tmath|\operatorname{M}_2(\Z)}} the additive identity is
: $0 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$
In the quaternions, 0 is the additive identity.
In the ring of functions from {{tmath|\R \to \R}}, the function mapping every number to 0 is the additive identity.
In the additive group of vectors in {{tmath|\R^n,}} the origin or zero vector is the additive identity.

Properties

=The additive identity is unique in a group=

Let {{math|(G, +)}} be a group and let {{math|0}} and {{math|0'}} in {{mvar|G}} both denote additive identities, so for any {{mvar|g}} in {{mvar|G}},

: $0+g = g = g+0, \qquad 0'+g = g = g+0'.$

It then follows from the above that

: ${\color{green}0'} = {\color{green}0'} + 0 = 0' + {\color{red}0} = {\color{red}0}.$

=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 {{mvar|s}} in {{mvar|S}}, {{math|1=s · 0 = 0}}. This follows because:

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

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

Let {{mvar|R}} be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let {{mvar|r}} be any element of {{mvar|R}}. Then

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

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