Additive Identity Property Of 0

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 whatever element x in the fix, 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 equally in groups and rings.

Elementary examples [edit]

Formal definition [edit]

Let Due north exist a group that is closed under the functioning of addition, denoted +. An additive identity for N, denoted e, is an element in Due north such that for any element north in N,

e + n = n = n + e.

Further examples [edit]

In a group, the additive identity is the identity element of the grouping, is ofttimes denoted 0, and is unique (run into below for proof).
A ring or field is a group nether the functioning of addition and thus these also have a unique condiment identity 0. This is defined to be unlike from the multiplicative identity one if the ring (or field) has more than 1 element. If the additive identity and the multiplicative identity are the same, then the ring is fiddling (proved below).
In the band Chiliad_m×n(R) of m by n matrices over a ring R, the additive identity is the nix matrix,^[i] denoted O or 0, and is the 1000 past north matrix whose entries consist entirely of the identity chemical element 0 in R. For example, in the 2×2 matrices over the integers M₂(Z) the additive identity is

$0={\begin{bmatrix}0&0\\0&0\cease{bmatrix}}$
In the quaternions, 0 is the condiment 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.

Properties [edit]

The condiment identity is unique in a group [edit]

Let (G, +) be a group and let 0 and 0' in G both announce additive identities, so for whatever yard in G,

0 + g = thousand = g + 0 and 0' + g = g = grand + 0'.

It then follows from the above that

0' = 0' + 0 = 0' + 0 = 0.

The additive identity annihilates ring elements [edit]

In a system with a multiplication operation that distributes over add-on, the additive identity is a multiplicative absorbing chemical element, meaning that for whatsoever s in S, s·0 = 0. This follows because:

{\brainstorm{aligned}southward\cdot 0&=s\cdot (0+0)=due south\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-southward\cdot 0\\\Rightarrow s\cdot 0&=0.\end{aligned}}

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

Let R exist a ring and suppose that the condiment identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let r be whatever chemical element of R. Then

r = r × i = r × 0 = 0

proving that R is petty, i.e. R = {0}. The contrapositive, that if R is non-picayune then 0 is not equal to ane, is therefore shown.

References [edit]

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

Bibliography [edit]

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

External links [edit]

Uniqueness of additive identity in a ring at PlanetMath.