The Additive identity
In maths the additive identity of a set which is kept with the operation of addition is an element which, when joined to any element x in the set, yields x. One of the most familiar additive identities is the number zero from elementary maths, but additive identities occur in other mathematical structures where addition is defined, like in groups and rings.
Elementary examples
The additive identity familiar from elementary maths is 0, denoted zero. For example,
5 + 0 = 5 = 0 + 5.
In the natural numbers N and all of its supersets (the integers Z, the rational numbers Q, the real numbers R, or the complex numbers C), the additive identity is zero. Thus for any one of these numbers n,
n + 0 = n = 0 + n.
Further examples
In a group the additive identity is the identity element of the group, is often denoted zero, and is unique.
A ring or field is a group under the operation of addition and thus these also have a unique additive identity zero. This is defined to be different from the multiplicative identity 1 if the ring has more than 1 element. If the additive identity and the multiplicative identity are the same, then the ring is trivial.
In the ring Mm×n(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element zero in R. For
Proofs
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 and 0' + g = g = g + 0'.
It follows from the above that
0 + (0') = (0') = (0') + 0 and 0' + (0) = (0) = (0) + 0'
which shows that 0 = 0'.
The additive and multiplicative identities are different in a non-trivial ring
Let R be a ring and suppose that the additive identity zero and the multiplicative identity one are equal, or 0 = 1. Let r be any element of R. Then
r = r × 1 = r × 0 = 0,
proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.
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 can be seen because s·0 = s·(0 + 0) = s·0 + s·0, so that, by cancellation s·0 = 0.
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