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Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?

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up vote 11 down vote accepted

Let $\mathfrak{p}$ be a prime ideal in $R$. Then $R/\mathfrak{p}$ is a finite integral domain, thus it is a field, hence $\mathfrak{p}$ is maximal.

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