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.
Formal definition.
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",
Properties.
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",
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", . 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.