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

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 $N$,
 * $$e+n = n = n+e.$$

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 $M_{m × n}(R)$ 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 $\operatorname{M}_2(\Z)$ 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 $\R \to \R$, 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.

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 $G$,
 * $$0+g = g = g+0, \qquad 0'+g = g = g+0'.$$

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 $s$ in $S$, $s · 0 = 0$. 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 $R$ 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


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

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.