Let $R$ be a commutative ring with unity such that for every minimal prime ideal $q$ of $R$ , $R/q$ is finite , then is it true that $R/P$ is finite for every prime ideal $P$ i.e. that $R$ is of $0$ Krull dimension with all residue fields finite ?

I know the claim is true for Noetherian rings but don't know what happens in general .


Let $P$ be any prime ideal in $R$. Then $P$ contains a minimal prime $Q$. There is a surjection of $R/Q$ onto $R/P$. So if $R/Q$ is finite, then so is $R/P$.

In particular the residue fields (quotients by maximal ideals) are finite.

Now, let $P$ be any prime ideal in $R$. The quotient $R/P$ is a domain with finitely many elements. Therefore it is a field! (Multiplication by a fixed nonzero element is injective on the set of nonzero elements, since the ring is a domain. By finiteness, the multiplication is also surjective.) Therefore every prime ideal in $R$ is maximal, and $R$ has Krull dimension $0$.


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