A maximal ideal in a commutative ring is prime

Let $𝑅$ be a commutative ring and $𝐼 ◃ 𝑅$ be a maximal ideal. Then $𝐼$ is a prime ideal. #m/thm/ring

Proof

$𝑅 / 𝐼$ for commutative $𝑅$ is a field iff $𝐼$ is maximal and $𝑅 / 𝐼$ for commutative $𝑅$ is an integral domain iff $𝐼$ is prime.

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