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.
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,
Further examples
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 identity1 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-nmatrices 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 the additive identity is
Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G,
It then follows from the above that
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 follows because:
The additive and multiplicative identities are different in a non-trivial ring
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
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.