Additive identity

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

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

5 + 0 = 5 = 0 + 5. {\displaystyle 5+0=5=0+5.}
* In the natural numbers ⁠ N {\displaystyle \mathbb {N} } ⁠ (if 0 is included), the integers ⁠ Z , {\displaystyle \mathbb {Z} ,} ⁠ the rational numbers ⁠ Q , {\displaystyle \mathbb {Q} ,} ⁠ the real numbers ⁠ R , {\displaystyle \mathbb {R} ,} ⁠ and the complex numbers ⁠ C , {\displaystyle \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 . {\displaystyle n+0=n=0+n.}

## Formal definition

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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 . {\displaystyle e+n=n=n+e.}

## Further examples

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* 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 Mm × 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 ⁠ M 2 ⁡ ( Z ) {\displaystyle \operatorname {M} _{2}(\mathbb {Z} )} ⁠ the additive identity is

0 = [ 0 0 0 0 ] {\displaystyle 0={\begin{bmatrix}0&0\\\0&0\end{bmatrix}}}
* In the quaternions, 0 is the additive identity.
* In the ring of functions from ⁠ R -> R {\displaystyle \mathbb {R} \to \mathbb {R} } ⁠, the function mapping every number to 0 is the additive identity.
* In the additive group of vectors in ⁠ R n , {\displaystyle \mathbb {R} ^{n},} ⁠ the origin or zero vector is the additive identity.

## Properties

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### The additive identity is unique in a group

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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 , 0 ′ + g = g = g + 0 ′ . {\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}

It then follows from the above that

0 ′ = 0 ′ + 0 = 0 ′ + 0 = 0 . {\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}0}.}

### The additive identity annihilates ring elements

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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:

s ⋅ 0 = s ⋅ ( 0 + 0 ) = s ⋅ 0 + s ⋅ 0 ⇒ s ⋅ 0 = s ⋅ 0 − s ⋅ 0 ⇒ s ⋅ 0 = 0. {\displaystyle {\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

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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 × 1 = r × 0 = 0 {\displaystyle 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.

## See also

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

## References

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

## Bibliography

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

## External links

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* Uniqueness of additive identity in a ring at PlanetMath.