Does the ring extension $\mathbb{Z}\subset\mathbb{Z}[\frac{1}{10}]$ satisfy the going up property?

Asked Feb 09 '23 at 13:32

Active Feb 09 '23 at 14:06

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Going up property: For a ring extension $A\subset B$, if for any prime ideals $p\subset p'$ of $A$ and prime ideal $P$ in $B$ that lies over $p$, there is a prime ideal $P'$ in $B$ such that $P\subset P'$ and $P'$ lies over $p'$.

I know that a chain of prime ideals in $\mathbb{Z}$ is:

$$(0)\subset (p)$$ for primes $p\in\mathbb{Z}$ and a chain of prime ideals in $\mathbb{Z}[\frac{1}{10}]$ is:

$$(0)\subset (p)$$ for $p\in\mathbb{Z}$ except $p=2,5$.

Then clearly the going up property is satisfied for chains where $p\neq 2,5$. But I am confused if it has to also hold for the other chains as well. In other words, the going up property is said to be satisfied if it holds for all chains of prime ideals?

Could someone kindly clarify this?

edited Feb 09 '23 at 14:06

asked Feb 09 '23 at 13:32

Anish Ray

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Please edit your post to recall the definition of the going up property. I think the answer will then become obvious. – Anne Bauval Feb 09 '23 at 13:40
@AnneBauval So, the answer is that the ring extension does satisfy the going up property, right? – Anish Ray Feb 09 '23 at 14:09
What is the reasoning which led you to this conclusion? – Anne Bauval Feb 09 '23 at 14:18
@AnneBauval I think I am a bit confused over the "any" before prime ideals in the definition. Could you kindly clarify this? – Anish Ray Feb 09 '23 at 14:45
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The "for any" means "for every" ... (such that ...). Formally: $\forall$ – Anne Bauval Feb 09 '23 at 14:46
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@AnneBauval Understood. Thank you for helping! – Anish Ray Feb 09 '23 at 14:48
For the chain of prime ideals $(0)\subset (2)$ in $\mathbb{Z}$ there is no prime ideal in $\mathbb{Z}[\frac{1}{10}]$ that lies over $(2)$, so this implies that the going up property is not satisfied. – Anish Ray Feb 09 '23 at 15:01
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I think the same. – Anne Bauval Feb 09 '23 at 15:04

Does the ring extension $\mathbb{Z}\subset\mathbb{Z}[\frac{1}{10}]$ satisfy the going up property?

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