Additive in terms of abelian? A lot of this is actually unecessary. For example if you have a semigroup (i.e. just a set with an associative binary operation ) then any identity would have to be unique, for if were both identities then because is an identity and since is an identity. Make sense? Now use this fact to prove that is unique, i.e. assume that are both inverses of and note that and are identities and so ...so