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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.

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

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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.

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