THC Science

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[edit]

The additive identity familiar from elementary mathematics is zero, denoted 0. For example,
$5+0=5=0+5.$
In the natural numbers ⁠ $\mathbb {N}$ ⁠ (if 0 is included), the integers ⁠ $\mathbb {Z} ,$ ⁠ the rational numbers ⁠ $\mathbb {Q} ,$ ⁠ the real numbers ⁠ $\mathbb {R} ,$ ⁠ and the complex numbers ⁠ $\mathbb {C} ,$ ⁠ 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[edit]

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[edit]

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 $M m \times n (R)$ of $m$ -by- $n$ matrices over a ring $R$ , the additive identity is the zero matrix,^[1] 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 ⁠ $\operatorname {M} _{2}(\mathbb {Z} )$ ⁠ 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 ⁠ $\mathbb {R} \to \mathbb {R}$ ⁠, the function mapping every number to 0 is the additive identity.
In the additive group of vectors in ⁠ $\mathbb {R} ^{n},$ ⁠ the origin or zero vector is the additive identity.

Properties[edit]

The additive identity is unique in a group[edit]

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[edit]

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 $S$ , $s \cdot 0 = 0$ . This follows because:

{\begin{aligned}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{aligned}}

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

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.

References[edit]

^ Weisstein, Eric W. "Additive Identity". mathworld.wolfram.com. Retrieved 2020-09-07.

Bibliography[edit]

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

External links[edit]

Uniqueness of additive identity in a ring at PlanetMath.

[1] Weisstein, Eric W. "Additive Identity". mathworld.wolfram.com. Retrieved 2020-09-07.

[1]

THC Science

Bringing Science to the Cannabis Conversation!

Cannabaceae