# Abstract Nonsense

## Localization (Pt. III)

Point of Post: This post is a continuation of this one.

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This tells us how to create a lot of ideals in $D^{-1}R$, sadly it doesn’t give us all the ideals! That said, there is a very nice correspondence between certain ideals (prime ideals) of $R$ and those of $D^{-1}R$. But first, we need a small lemma

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Lemma: Let $R$ be a commutative unital ring, and $D$ a unital multiplicative subset. Then, for any ideal $\mathfrak{a}$ of $R$, one has that $\langle\ell_D(\mathfrak{a})\rangle=D^{-1}R$ if and only if  $\mathfrak{a}\cap D\ne\varnothing$.

Proof: If $d\in\mathfrak{a}\cap D$ then we obviously have, by the above characterization of $\langle \ell_D(\mathfrak{a})\rangle$ that $\frac{d}{d}=1\in\langle \ell_D(\mathfrak{a})\rangle$ and so evidently $\langle\ell_D(\mathfrak{a})\rangle=D^{-1}R$. Conversely, if $\langle \ell_D(\mathfrak{a})\rangle=D^{-1}R$ then for some $a\in\mathfrak{a}$ and $d\in D$ we have that $\frac{a}{d}=\frac{1}{1}$ and so $d'a=dd'$. But, clearly $dd'\in D$ since $D$ is multiplicative, but $dd'\in\mathfrak{a}$ since $\mathfrak{a}$ is an ideal and $dd'=d'a$. Thus, $d'\in\mathfrak{a}\cap D$. $\blacksquare$

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With this we are able to prove the following correspondence between $\text{Spec}(D^{-1}R)$ and $\text{Spec}(R)$

Theorem: Let $R$ be a commutative unital ring and $D$ a unitally multiplicatively closed subset. Then, for $\mathfrak{p}\in\text{Spec}(R)$, one has that $\ell_D(\mathfrak{p})\in\text{Spec}(D^{-1}R)$ if and only if $\mathfrak{p}\cap D=\varnothing$.

Proof: Evidently if $\mathfrak{p}\cap D=\varnothing$ then by the above theorem we have that $\langle \ell_D(\mathfrak{p})\rangle=D^{-1}R$ and so is not proper, and so by definition, not prime. Conversely, suppose that $\mathfrak{a}\cap D=\varnothing$. Let $\displaystyle \frac{a}{d_1},\frac{a'}{d_2}\in D^{-1}R$ be such that $\frac{aa'}{d_1d_2}=\frac{p}{d_3}\in\langle \ell_D(\mathfrak{p})\rangle$. We see then that there exists $d_4\in D$ such that $d_4aa'd_3=d_1d_2pd_4$. Note then that the right hand side of this formula is in $\mathfrak{p}$ and so $d_4d_3aa'\in\mathfrak{p}$.  Now, we assumed that $\mathfrak{p}\cap D=\varnothing$, and so $d_4d_3\notin\mathfrak{p}$ and thus since $\mathfrak{p}$ is prime we have that $a\in\mathfrak{p}$ or $a'\in\mathfrak{p}$. Thus, $\frac{a}{d_1}\in\langle \ell_D(\mathfrak{p})\rangle$ or $\frac{a'}{d_2}\in\langle \ell_D(\mathfrak{p})\rangle$. The conclusion follows. $\blacksquare$

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The last piece of the puzzle for how $\text{Spec}(D^{-1}R)$ and $\text{Spec}(R)$ are related is given in the following theorem:

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Theorem: Let $R$ be a commutative unital ring, $D$ a unital multiplicative subset. Then, if $\mathfrak{p}$ is a prime ideal in $R$ disjoint from $D$ then $\ell_D^{-1}\left(\langle \ell_D(\mathfrak{p})\rangle\right)=\mathfrak{p}$.

Proof: We trivially have that $\mathfrak{p}\subseteq\ell_D^{-1}\left(\langle\ell_D(\mathfrak{p})\rangle\right)$ and so it suffices to prove the reverse inclusion. Indeed, let $\frac{r}{1}\in\langle \ell_D(\mathfrak{p})\rangle$, then there exists $d,d'\in D$ and $p\in\mathfrak{p}$ with $dd'r=d'p$. Now, the right hand side of this equation is in $\mathfrak{p}$ and so obviously the left hand side is as well. But, since $\mathfrak{p}$ is prime this tells us that either $dd'\in\mathfrak{p}$ or $r\in\mathfrak{p}$, but since we assumed that the former can’t happen, we may conclude the latter. Since $r$ was arbitrary the conclusion follows. $\blacksquare$

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From this we get the conclusive relationship between the spectrum of a ring and the spectrum of its localization at a unital multiplicative subset:

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Theorem: The map $\left\{\mathfrak{p}\in\text{Spec}(R):\mathfrak{p}\cap D=\varnothing\right\}\to\text{Spec}(D^{-1}R)$ given by $\mathfrak{p}\mapsto \langle \ell_D(\mathfrak{p})\rangle$ is a bijection.

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Localizing at a Prime Ideal

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We now discuss what is perhaps one of the most important cases of localizations, localization of a ring at a prime ideal, usually denoted $R_\mathfrak{p}$. This terminology is slightly confusing since it would seem to imply the multiplicative subset we are localizing at is the prime ideal itself. But, this is impossible since no prime ideal contains $1$. In fact, we define for a commutative unital ring $R$ and a prime ideal $\mathfrak{p}\in\text{Spec}(R)$  the localization at the prime ideal $\mathfrak{p}$ denoted $R_\mathfrak{p}$ to be equal to $D^{-1}R$ where $D=R-\mathfrak{p}$. Perhaps one of the most instructive examples comes from taking $R=\mathbb{Z}$ and $\mathfrak{p}=(p)$ for some prime $p$. We see then by inspection that $\mathbb{Z}_{(p)}$ is precisely the reduced elements of $\mathbb{Q}$ which have denominators coprime to $p$.

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For now all I want to say about localization at a prime ideal, is that they create what are called (unsurprisingly) local rings. Local rings are those with a unique maximal ideal. We know stupid examples, like simple rings such as fields or matrices over fields, but we have never seen an example of a local ring which has a non-trivial ideal theory. Well, the localization of a ring at a prime ideal provides such examples. Indeed:

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Theorem: Let $R$ be a commutative unital ring and $D$ a unital multiplicative subset. Then, if $\mathfrak{p}$ is a prime ideal of $R$ the ring $R_\mathfrak{p}$ has the unique maximal ideal $\mathfrak{p}R_\mathfrak{p}$.

Proof: We know from the above analysis of $\text{Spec}\left(R_\mathfrak{p}\right)$ that any maximal ideal of $R_\mathfrak{p}$ will be the image of a maximal-by-containment ideal in $R$ disjoint from $R-\mathfrak{p}$, and so a maximal-by-containment ideal contained in $\mathfrak{p}$. Since clearly the only such maximal-by-containment ideal in the ideal sitting inside $\mathfrak{p}$ itself, we may conclude that the only element of $\text{MaxSpec}(R_\mathfrak{p})$ is $\langle \ell_D(\mathfrak{p})\rangle=\mathfrak{p}R_\mathfrak{p}$. $\blacksquare$

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References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.