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 in the set, yields . 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.
5+0=5=0+5.
n+0=n=0+n.
Let be a group that is closed under the operation of addition, denoted +. An additive identity for, denoted, is an element in such that for any element in,
e+n=n=n+e.
0=\begin{bmatrix}0&0\ 0&0\end{bmatrix}
Let be a group and let and in both denote additive identities, so for any in,
0+g=g=g+0, 0'+g=g=g+0'.
It then follows from the above that
{\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 in, . This follows because:
\begin{align} s ⋅ 0&=s ⋅ (0+0)=s ⋅ 0+s ⋅ 0\\ ⇒ s ⋅ 0&=s ⋅ 0-s ⋅ 0\\ ⇒ s ⋅ 0&=0. \end{align}
Let be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let be any element of . Then
r=r x 1=r x 0=0
proving that is trivial, i.e. The contrapositive, that if is non-trivial then 0 is not equal to 1, is therefore shown.