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.

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

  ${\displaystyle 5+0=5=0+5.}$

• In the natural numbers ⁠${\displaystyle \mathbb {N} }$⁠ (if 0 is included), the integers ⁠${\displaystyle \mathbb {Z} ,}$⁠ the rational numbers ⁠${\displaystyle \mathbb {Q} ,}$⁠ the real numbers ⁠${\displaystyle \mathbb {R} ,}$⁠ and the complex numbers ⁠${\displaystyle \mathbb {C} ,}$⁠ the additive identity is 0. This says that for a number n belonging to any of these sets,

  ${\displaystyle n+0=n=0+n.}$

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,

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

• 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,^[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 ⁠${\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}$⁠ the additive identity is

  ${\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}$

• In the quaternions, 0 is the additive identity.
• In the ring of functions from ⁠${\displaystyle \mathbb {R} \to \mathbb {R} }$⁠, the function mapping every number to 0 is the additive identity.
• In the additive group of vectors in ⁠${\displaystyle \mathbb {R} ^{n},}$⁠ the origin or zero vector is the additive identity.

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

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

It then follows from the above that

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

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:

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

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

${\displaystyle r=r\times 1=r\times 0=0}$

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.

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

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

• Uniqueness of additive identity in a ring at PlanetMath.