Every nontrivial commutative ring has a maximal ideal

Every nontrivial commutative ring has a maximal ideal: If $R$ is a commutative ring with $1$ and $1\neq 0$, then there exists a maximal ideal $\mathfrak m\lhd R$.

This follows from the existence theorem for maximal ideals , whose proof applies Zorn's Lemma to the set of proper ideals of a commutative ring with $1$ to produce a maximal ideal .