About condition of The Incomparability theorem

Question

The Incomparability theorem: Let a commutative ring S is integral over its subring R. Suppose Q and Q' be two prime ideals of S with $ Q \subseteq Q' $ and $ Q \cap R = Q' \cap R $ then Q=Q'.

I have proved it. My question is about the condition " S is integral over R " . Can this condition dropped?

Jim · Accepted Answer · 2020-02-17T04:33:50.553

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Note that $(x)$ and $(x, y)$ are prime ideals of $k[x, y]$ where $k$ is a field, and $(x) \cap k = \{0\} = (x, y) \cap k$.

edited Feb 17 '20 at 04:33

answered Feb 17 '20 at 04:20

Jim

31,547

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I think 2nd one should be taken (x, y) otherwise $ Q \subseteq Q' $ does not hold – Pradip Feb 17 '20 at 04:32
@Pradip: Whoops, yea, I missed that hypothesis. – Jim Feb 17 '20 at 04:33

About condition of The Incomparability theorem

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