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I have to answer the following question:

Compute the prime ideals of the localization of $\mathbb Z$ given by the multiplicative set of the powers of $n$ (such that $n>1$) and find the nilradical of this ring.

I have thought this:

Let $\mathbb Z _n$ be such localization. I will use the following theorem:

Let $S$ be a multiplicative set. There exists a bijection between the prime ideals $P$ of $A$ such that $P\cap S = \emptyset$ and the prime ideals of $A_S$.

Let $P=(p)$ (with $p$ prime number) be a prime ideal of $\mathbb Z$, then it is bijective to a prime ideal of $\mathbb Z_n$ only if $n\notin (p)$.

Is this reasoning right? How can I approach the part about the nilradical?

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  • $\begingroup$ Any localization of a reduced ring is reduced $\endgroup$
    – D_S
    Commented Jan 2, 2018 at 4:10

3 Answers 3

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Theorem: if $\varphi$ is a locatization morphism $A\to A_S$, then $spec \varphi$ is a homeomorphism from $spec(A_S)$ onto the subspace $\{p\in SpecA; p\cap S= \phi$ of $spec A \}$. (Q.Liu, Algebraic geometry, p28 )

Theorem: Let $X$ affine scheme. $X$ is a reduced affine scheme $\Leftrightarrow \sqrt{(0)} =(0) $. (Robin Hartshorn, Algebraic geometry, p82)

We know $\mathbb Z$ is an integral domain, then $\mathbb Z_S$ is also an integral domain. Hence $X=spec \mathbb Z_S$ is an integral affine scheme, then $X=spec \mathbb Z_S$ is reduced and irreducible. i.e. $\sqrt{(0)} =(0)$

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Your might translate your answer to the first part, in a more explicit way, as

The bijection maps the primes non-divisors of $n$ to the prime ideals of $\mathbf Z_n$.

For the nilradical question, remember the localisation of an integral domain is an integral domain.

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  • $\begingroup$ Non-prime divisors of $n$? You mean prime non-divisors of $n$? As written it sounds like it is saying the composite divisors of $n$. $\endgroup$ Commented Jan 2, 2018 at 6:02
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    $\begingroup$ @ZachTeitler: That's right. For my excuse, 'twas New Year's Day. I've fixed it. Thanks for pointing the ambiguity, and happy new year! $\endgroup$
    – Bernard
    Commented Jan 2, 2018 at 11:05
  • $\begingroup$ Don't forget the prime ideal $(0).$ $\endgroup$ Commented Feb 9, 2023 at 12:40
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Your proof for the first part looks fine, although you might be a little more explicit about enumerating the prime ideals of $\mathbb{Z}_n$.

For the second part, it might help to remember that the nilradical of a commutative ring with unity is the intersection of all the prime ideals.

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