Theorem. Let $I$ be an ideal in a commutative ring $R$ with identity. Then $I$ is a maximal ideal if and only if the quotient ring $R/I$ is a field.
Above Theorem is very famous theorem. But The Theorem is hold under condition $R$ is a commutative ring with identity.
I want to know whether this theorem holds under condition $R$ is a nontrivial commutative ring.