Example that $R/I$ is not field where $R$ is a commutative ring and $I$ is maximal ideal.

Question

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.

Answer 1

$2\mathbb{Z}/4\mathbb{Z}$ is not a field because it has zero multiplication.

Answer 2

Let $R=\mathbb Z_2$ and define the operation $\circ$ on $R$ by $x\circ y=0$. Then $(R,+,\circ)$ is a commutative ring (without identity) and $I=\{0\}$ is a maximal ideal, but $R/I\cong R$ is not a field.