Yeah, as an example this is a simple corollary of the following result which gets used a lot: the quotient of a (unital, commutative) ring by a maximal ideal is

Yeah, as an example this is a simple corollary of the following result which gets used a lot: the quotient of a (unital, commutative) ring by a maximal ideal is always a field. If your ring has only two ideals then the trivial ideal is maximal, and thus your ring is already a field! The more you know, the more "shortcuts" you start seeing, I guess.