Let R be a commutative ring with identity 1 \neq 0. Prove that R is a field iff the zero ideal is the maximal ideal of R.
Proof.
“\Rightarrow“: Assume R is a field. Then we saw here that 0 and R are the only ideals of R. Since a maximal ideal can’t be R itself, it must be that the maximal ideal is the zero ideal.
“\Leftarrow“: Assume that the zero ideal of R is a maximal ideal. Since it is possible to have more than one maximal ideal, we need to show that it is not the case here. Assume there is an ideal I in R such that it doesn’t contain the zero ideal. By definition of an ideal, zero is always contained in an ideal. So that is a contradiction. This means that R has two ideals: the zero ideal and R itself. But that is already the definition of a field, so R is a field.