The Stacks project

Lemma 10.50.17. Let $A$ be a valuation ring. Ideals in $A$ correspond $1 - 1$ with ideals of $\Gamma $. This bijection is inclusion preserving, and maps prime ideals to prime ideals.

Proof. Omitted. $\square$


Comments (3)

Comment #8760 by Zhenhua Wu on

the zero prime ideal of doesn't correspond to any ideal of because by the definition here we don't allow zero ideal in , unless you allow to be an ideal of .

Comment #8874 by Zhenhua Wu on

Sorry for the last comment. Actually from the definition of ideals of we can see that and can all be ideals. They correspond to and respectively. But is not a prime ideal, so we shouldn't allow to be a prime ideal of .

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  • 3 comment(s) on Section 10.50: Valuation rings

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