I was wondering if an Ideal in a ring can ever be the null set. The definition of an Ideal $I$ is that it is a subset of the ring $R$ such that:
1)It is an abelian group under "addition" (I put it in inverted commas, since the operation is not strictly addition but rather additive notation is used)
2) $\forall r \in R$, $a \in I, a.r, r.a \in I $
So under this definition we can have the zero ring as an ideal since it trivially satisfies all the definitions, but I can't apply the same reasoning for $\emptyset$ So is $\emptyset$ ever an ideal of any ring?
Usually an ideal also has to be an additive subgroup of the additive structure of the ring.
Groups are never empty (they at least contain an identity), so ideals cannot be empty.