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

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I have to admit that a ring denoted by $P$ looks very strange to me. –  user26857 Jan 5 '13 at 16:55
    
@YACP: Maybe it is Cyrillic. –  azimut Sep 13 '13 at 15:15
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up vote 8 down vote accepted

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

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