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, 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 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.
The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.