# Rings of Fractions and Localisation

Is perhaps the most important technical tools in commutative algebra. In this post we are covering definitions and simple properties. Also we restrict ourselves into ring theories and no further than that. Throughout, we let \(A\) be a commutative ring. With extra effort we can also make it to non-commutative rings for some results but we are not doing that here.

In fact the construction of \(\mathbb{Q}\) from \(\mathbb{Z}\) has already been an example.
For any \(a \in \mathbb{Q}\), we have
some \(m,n \in \mathbb{Z}\) with \(n \neq 0\) such that \(a = \frac{m}{n}\). As a matter of notation
we may also say an ordered pair \((m,n)\) determines \(a\). Two ordered pairs \((m,n)\) and \((m',n')\) are *equivalent*
if and only if \[
mn'-m'n=0.
\] But we are only using the ring structure of \(\mathbb{Z}\). So it is natural to think
whether it is possible to generalize this process to all rings. But we
are also using the fact that \(\mathbb{Z}\) is an entire ring (or
alternatively integral domain, they mean the same thing). However there
is a way to generalize it. \(\def\mfk{\mathfrak}\)

## Multiplicatively closed subset

(Definition 1)Amultiplicatively closed subset\(S \subset A\) is a set that \(1 \in S\) and if \(x,y \in S\), then \(xy \in S\).

For example, for \(\mathbb{Z}\) we have a multiplicatively closed subset \[ \{1,2,4,8,\cdots\} \subset \mathbb{Z}. \] We can also insert \(0\) here but it may produce some bad result. If \(S\) is also an ideal then we must have \(S=A\) so this is not very interesting. However the complement is interesting.

(Proposition 1)Suppose \(A\) is a commutative ring such that \(1 \neq 0\). Let \(S\) be a multiplicatively closed set that does not contain \(0\). Let \(\mfk{p}\) be the maximal element of ideals contained in \(A \setminus S\), then \(\mfk{p}\) is prime.

*Proof.* Recall that \(\mfk{p}\) is prime if for any \(x,y \in A\) such that \(xy \in \mfk{p}\), we have \(x \in \mfk{p}\) or \(y \in \mfk{p}\). But now we fix \(x,y \in \mfk{p}^c\). Note we have a
strictly bigger ideal \(\mfk{q}_1=\mfk{p}+Ax\). Since \(\mfk{p}\) is maximal in the ideals
contained in \(A \setminus S\), we see
\[
\mfk{q}_1 \cap S \neq \varnothing.
\] Therefore there exist some \(a \in
A\) and \(p \in \mfk{p}\) such
that \[
p+ax \in S.
\] Also, \(\mfk{q}_2=\mfk{p}+Ay\) has nontrivial
intersection with \(S\) (due to the
maximality of \(\mfk{p}\)), there exist
some \(a' \in A\) and \(p' \in \mfk{p}\) such that \[
p' + a'y \in S.
\] Since \(S\) is closed under
multiplication, we have \[
(p+ax)(p'+a'y) = pp'+p'ax+pa'y+aa'xy \in S.
\] But since \(\mfk{p}\) is an
ideal, we see \(pp'+p'ax+pa'y \in
\mfk{p}\). Therefore we must have \(xy
\notin \mfk{p}\) since if not, \((p+ax)(p'+a'y) \in \mfk{p}\), which
gives \(\mfk{p} \cap S \neq
\varnothing\), and this is impossible. \(\square\)

As a corollary, for an ideal \(\mfk{p} \subset A\), if \(A \setminus \mfk{p}\) is multiplicatively closed, then \(\mfk{p}\) is prime. Conversely, if we are given a prime ideal \(\mfk{p}\), then we also get a multiplicatively closed subset.

(Proposition 2)If \(\mfk{p}\) is a prime ideal of \(A\), then \(S = A \setminus \mfk{p}\) is multiplicatively closed.

*Proof.* First \(1 \in S\)
since \(\mfk{p} \neq A\). On the other
hand, if \(x,y \in S\) we see \(xy \in S\) since \(\mfk{p}\) is prime. \(\square\)

## Ring of fractions of a ring

We define a equivalence relation on \(A \times S\) as follows: \[ (a,s) \sim (b,t) \iff \exists u \in S, (at-bs)u=0. \]

(Proposition 3)\(\sim\) is an equivalence relation.

*Proof.* Since \((as-as)1=0\)
while \(1 \in S\), we see \((a,s) \sim (a,s)\). For being symmetric,
note that \[
(at-bs)u=0 \implies (bs-at)u=0 \implies (b,t) \sim (a,s).
\] Finally, to show that it is transitive, suppose \((a,s) \sim (b,t)\) and \((b,t) \sim (c,u)\). There exist \(u,v \in S\) such that \[
(at-bs)v=(bu-ct)w=0.
\] This gives \(bsv=atv\) and
\(buw = ctw\), which implies \[
bsvuw=atvuw=ctwsv \implies (au-cs)tvw =0.
\] But \(tvw \in S\) since \(t,v,w \in S\) and \(S\) is multiplicatively closed. Hence \[
[(a,s) \sim (b,t)] \land [(b,t) \sim (c,u)] \implies (a,s) \sim (c,u).
\] \(\square\)

Let \(a/s\) denote the equivalence class of \((a,s)\). Let \(S^{-1}A\) denote the set of equivalence classes (it is not a good idea to write \(A/S\) as it may coincide with the notation of factor group), and we put a ring structure on \(S^{-1}A\) as follows: \[ (a/s)+(b/t)=(at+bs)/st, \\ (a/s)(b/t)=ab/st. \] There is no difference between this one and the one in elementary algebra. But first of all we need to show that \(S^{-1}A\) indeed form a ring.

(Proposition 4)The addition and multiplication are well defined. Further, \(S^{-1}A\) is a commutative ring with identity.

*Proof.* Suppose \((a,s) \sim
(a',s')\) and \((b,t) \sim
(b',t')\) we need to show that \[
(a/s)+(b/t)=(a'/s')+(b'/t')
\] or \[
(at+bs)/st = (a't'+b's')/s't'.
\] There exists \(u,v \in S\)
such that \[
(as'-a's)u=0 \quad (bt'-b't)v=0.
\] If we multiply the first equation by \(vtt'\) and second equation by \(uss'\), we see \[
as'uvtt'-a'suvtt'+bt'vuss'-b'tvuss'=[(at)s't'+(bs)s't'-(a't')st-(b's')st]uv,
\] which is exactly what we want.

On the other hand, we need to show that \[ ab/st = a'b'/s't'. \] That is, \[ \exists y \in S,(abs't'-a'b'st)y=0. \] Again, we have \[ (as'-a's)u=(as'-a's)uvbt'=(abs't'-a'bst')uv=0, \\ (bt'-b't)v=(bt'-b't)vua's=(a'bst'-a'b'st)uv=0. \] Hence \[ (abs't'-a'bst')uv+(a'bst'-a'b'st)uv=(abs't'-a'b'st)uv=0. \] Since \(uv \in S\), we are done.

Next we show that \(S^{-1}A\) has a ring structure. If \(0 \in S\), then \(S^{-1}A\) contains exactly one element \(0/1\) since in this case, all pairs are equivalent: \[ (at-bs)0=0. \] We therefore only discuss the case when \(0 \notin S\). First \(0/1\) is the zero element with respect to addition since \[ 0/1+a/s = (0s+1a)/1s = a/s. \] On the other hand, we have the inverse \(-a/s\): \[ -a/s+a/s = (-as+as)/ss=0/ss=0/1. \] \(1/1\) is the unit with respect to multiplication: \[ (1/1)(a/s)=1a/1s=a/s. \] Multiplication is associative since \[ [(a/s)(b/t)](c/u)=(ab/st)(c/u)=abc/stu. \\ (a/s)[(b/t)(c/u)]=(a/s)(bc/tu)=abc/stu. \] Multiplication is commutative since \[ ab/st+(-ba)/st=(abst-bast)/s^2t^2=0. \] Finally distributivity. \[ (a/s+b/t)(c/u)=(c/u)(a/s+b/t)=[(at+bs)/st](c/u)=(act+bcs)/stu \\ (a/s)(c/u)+(b/t)(c/u)=ac/su+bc/tu=(actu+bcsu)/stu^2=(act/bcs)/stu \] Note \(ab/cb=a/c\) since \((abc-abc)1=0\). \(\square\) \(\def\mb{\mathbb}\)

### Cases and examples

First we consider the case when \(A\) is entire. If \(0 \in S\), then \(S^{-1}A\) is trivial, which is not so interesting. However, provided that \(0 \notin S\), we get some well-behaved result:

(Proposition 5)Let \(A\) be an entire ring, and let \(S\) be a multiplicatively closed subset of \(A\) that does not contain \(0\), then the natural map \[ \begin{aligned} \varphi_S: A &\to S^{-1}A \\ x &\mapsto x/1 \end{aligned} \] is injective. Therefore it can be considered as a natural inclusion. Further, every element of \(\varphi_S(S)\) is invertible.

*Proof.* Indeed, if \(x/1=0/1\), then there exists \(s \in S\) such that \(xs=0\). Since \(A\) is entire and \(s \neq 0\), we see \(x=0\), hence \(\varphi_S\) is entire. For \(s \in S\), we see \(\varphi_S(s)=s/1\). However \((1/s)\varphi_S(s)=(1/s)(s/1)=s/s=1\). \(\square\)

Note since \(A\) is entire we can
also conclude that \(S^{-1}A\) is
entire. As a word of warning, the ring homomorphism \(\varphi_S\) is *not* in general
injective since, for example, when \(0 \in
S\), this map is the zero.

If we go further, making \(S\) contain all non-zero element, we have:

(Proposition 6)If \(A\) is entire and \(S\) contains all non-zero elements of \(A\), then \(S^{-1}A\) is a field, called thequotient fieldor thefield of fractions.

*Proof.* First we need to show that \(S^{-1}A\) is entire. Suppose \((a/s)(b/t)=ab/st =0/1\) but \(a/s \neq 0/1\), we see however \[
ab/st=0/1 \implies \exists u \in S, (ab-0)u=0 \implies ab=0.
\] Since \(A\) is entire, \(b\) has to be \(0\), which implies \(b/t=0/1\). Second, if \(a/s \neq 0/1\), we see \(a \neq 0\) and therefore is in \(S\), hence we've found the inverse \((a/s)^{-1}=s/a\). \(\square\)

In this case we can identify \(A\) as a subset of \(S^{-1}A\) and write \(a/s=s^{-1}a\).

Let \(A\) be a commutative ring, an
let \(S\) be the set of invertible
elements of \(A\). If \(u \in S\), then there exists some \(v \in S\) such that \(uv=1\). We see \(1 \in S\) and if \(a,b \in S\), we have \(ab \in S\) since \(ab\) has an inverse as well. This set is
frequently denoted by \(A^\ast\), and
is called the group of **invertible** elements of \(A\). For example for \(\mb{Z}\) we see \(\mb{Z}^\ast\) consists of \(-1\) and \(1\). If \(A\) is a field, then \(A^\ast\) is the multiplicative group of
non-zero elements of \(A\). For example
\(\mb{Q}^\ast\) is the set of all
rational numbers without \(0\). For
\(A^\ast\) we have

If \(A\) is a field, then \((A^\ast)^{-1}A \simeq A\).

*Proof.* Define \[
\begin{aligned}
\varphi_S:A &\to (A^\ast)^{-1}A \\
x &\mapsto x/1.
\end{aligned}
\] Then as we have already shown, \(\varphi_S\) is injective. Secondly we show
that \(\varphi_S\) is surjective. For
any \(a/s \in (A^\ast)^{-1}A\), we see
\(as^{-1}/1 = a/s\). Therefore \(\varphi_S(as^{-1})=a/s\) as is shown. \(\square\)

Now let's see a concrete example. If \(A\) is entire, then the polynomial ring
\(A[X]\) is entire. If \(K = S^{-1}A\) is the quotient field of
\(A\), we can denote the quotient field
of \(A[X]\) as \(K(X)\). Elements in \(K(X)\) can be naturally called
**rational polynomials**, and can be written as \(f(X)/g(X)\) where \(f,g \in A[X]\). For \(b \in K\), we say a rational function \(f/g\) is **defined** at \(b\) if \(g(b)
\neq 0\). Naturally this process can be generalized to
polynomials of \(n\) variables.

## Local ring and localization

We say a commutative ring \(A\) is
local if it has a unique maximal ideal. Let \(\mfk{p}\) be a prime ideal of \(A\), and \(S = A
\setminus \mfk{p}\), then \(A_{\mfk{p}}=S^{-1}A\) is called the
**local ring of \(A\) at \(\mfk{p}\)**. Alternatively, we say
the process of passing from \(A\) to
\(A_\mfk{p}\) is *localization*
at \(\mfk{p}\). You will see it makes
sense to call it localization:

(Proposition 7)\(A_\mfk{p}\) is local. Precisely, the unique maximal ideal is \[ I=\mfk{p}A_\mfk{p}=\{a/s:a \in \mfk{p},s \in S\}. \] Note \(I\) is indeed equal to \(\mfk{p}A_\mfk{p}\).

*Proof.* First we show that \(I\) is an ideal. For \(b/t \in A_\mfk{p}\) and \(a/s \in I\), we see \[
(b/t)(a/s)=ba/ts \in A_\mfk{p}
\] since \(a \in \mfk{p}\)
implies \(ba \in \mfk{p}\). Next we
show that \(I\) is maximal, which is
equivalent to show that \(A_\mfk{p}/I\)
is a field. For \(b/t \notin I\), we
have \(b \in S\), hence it is legit to
write \(t/b\). This gives \[
(b/t+I)(t/b+I)=1/1+I.
\] Hence we have found the inverse.

Finally we show that \(I\) is the unique maximal ideal. Let \(J\) be another maximal ideal. Suppose \(J \neq I\), then we can pick \(m/n \in J \setminus I\). This gives \(m \in S\) since if not \(m \in \mfk{p}\) and then \(m/n \in I\). But for \(n/m \in A_\mfk{p}\) we have \[ (m/n)(n/m)=1/1 \in J. \] This forces \(J\) to be \(A_\mfk{p}\) itself, contradicting the assumption that \(J\) is a maximal ideal. Hence \(I\) is unique. \(\square\)

### Example

Let \(p\) be a prime number, and we take \(A=\mb{Z}\) and \(\mfk{p}=p\mb{Z}\). We now try to determine what do \(A_\mfk{p}\) and \(\mfk{p}A_\mfk{p}\) look like. First \(S = A \setminus \mfk{p}\) is the set of all entire numbers prime to \(p\). Therefore \(A_\mfk{p}\) can be considered as the ring of all rational numbers \(m/n\) where \(n\) is prime to \(p\), and \(\mfk{p}A_\mfk{p}\) can be considered as the set of all rational numbers \(kp/n\) where \(k \in \mb{Z}\) and \(n\) is prime to \(p\).

\(\mb{Z}\) is the simplest example of ring and \(p\mb{Z}\) is the simplest example of prime ideal. And \(A_\mfk{p}\) in this case shows what does localization do: \(A\) is 'expanded' with respect to \(\mfk{p}\). Every member of \(A_\mfk{p}\) is related to \(\mfk{p}\), and the maximal ideal is determined by \(\mfk{p}\).

Let \(k\) be a infinite field. Let
\(A=k[x_1,\cdots,x_n]\) where \(x_i\) are independent indeterminates, \(\mfk{p}\) a prime ideal in \(A\). Then \(A_\mfk{p}\) is the ring of all rational
functions \(f/g\) where \(g \notin \mfk{p}\). We have already defined
rational functions. But we can go further and demonstrate the prototype
of the local rings which arise in algebraic geometry. Let \(V\) be the variety defined by \(\mfk{p}\), that is, \[
V=\{x=(x_1,x_2,\cdots,x_n) \in k^n:\forall f \in \mfk{p}, f(x)=0\}.
\] Then what about \(A_\mfk{p}\)? We see since for \(f/g \in A_\mfk{p}\) we have \(g \notin \mfk{p}\), therefore for \(g(x)\) is not equal to \(0\) almost everywhere on \(V\). That is, \(A_\mfk{p}\) can be identified with the ring
of all rational functions on \(k^n\)
which are defined at *almost all* points of \(V\). We call this the local ring of \(k^n\) **along the variety**
\(V\).

## Universal property

Let \(A\) be a ring and \(S^{-1}A\) a ring of fractions, then we shall see that \(\varphi_S:S \to S^{-1}A\) has a universal property.

(Proposition 8)Let \(g:A \to B\) be a ring homomorphism such that \(g(s)\) is invertible in \(B\) for all \(s \in S\), then there exists a unique homomorphism \(h:S^{-1}A \to B\) such that \(g = h \circ \varphi_S\).

*Proof.* For \(a/s \in
S^{-1}A\), define \(h(a/s)=g(a)g(s)^{-1}\). It looks immediate
but we shall show that this is what we are looking for and is
unique.

Firstly we need to show that it is well defined. Suppose \(a/s=a'/s'\), then there exists some \(u \in S\) such that \[ (as'-a's)u=0. \] Applying \(g\) on both side yields \[ (g(a)g(s')-g(a')g(s))g(u)=0. \] Since \(g(x)\) is invertible for all \(s \in S\), we therefore get \[ g(a)g(s)^{-1}=g(a')g(s')^{-1}. \] It is a homomorphism since \[ \begin{aligned} h[(a/s)(a'/s')]&=g(a)g(a')g(s)^{-1}g(s')^{-1} \\ h(a/s)h(a'/s')&=g(a)g(s)^{-1}g(a')g(s')^{-1}, \end{aligned} \] and \[ h(a/s+a'/s')=h((as'+a's)/ss')=g(as'+a's)g(ss')^{-1} \\ h(a/s)+h(a'/s')=g(a)g(s)^{-1}+g(a')g(s')^{-1} \] they are equal since \[ \begin{aligned} g(as'+a's)g(ss')^{-1}&=g(as')g(ss')^{-1}+g(a's)g(ss')^{-1} \\ &=g(a)g(s')g(s)^{-1}g(s')^{-1}+g(a')g(s)g(s)^{-1}g(s')^{-1} \\ &=g(a)g(s)^{-1}+g(a')g(s')^{-1}. \end{aligned} \] Next we show that \(g=h \circ \varphi_S\). For \(a \in A\), we have \[ h(\varphi_S(a))=h(a/1)=g(a)g(1)^{-1}=g(a). \] Finally we show that \(h\) is unique. Let \(h'\) be a homomorphism satisfying the condition, then for \(a \in A\) we have \[ h'(a/1)=h'(\varphi_S(a))=g(a). \] For \(s \in S\), we also have \[ h'(1/s)=h'((s/1)^{-1})=h'(\varphi_S(s)^{-1})=h'(\varphi_S(s))^{-1}=g(s)^{-1}. \] Since \(a/s = (a/1)(1/s)\) for all \(a/s \in S^{-1}A\), we get \[ h'(a/s)=h'((a/1)(1/s))=g(a)g(s)^{-1}. \] That is, \(h'\) (or \(h\)) is totally determined by \(g\). \(\square\)

Let's restate it in the language of category theory (you can skip it if you have no idea what it is now). Let \(\mfk{C}\) be the category whose objects are ring-homomorphisms \[ f:A \to B \] such that \(f(s)\) is invertible for all \(s \in S\). Then according to proposition 5, \(\varphi_S\) is an object of \(\mfk{C}\). For two objects \(f:A \to B\) and \(f':A \to B'\), a morphism \(g \in \operatorname{Mor}(f,f')\) is a homomorphism \[ g:B \to B' \] such that \(f'=g \circ f\). So here comes the question: what is the position of \(\varphi_S\)?

Let \(\mfk{A}\) be a category. an
object \(P\) of \(\mfk{A}\) is called **universally
attracting** if there exists a unique morphism of each object of
\(\mfk{A}\) into \(P\), an is called **universally
repelling** if for every object of \(\mfk{A}\) there exists a unique morphism of
\(P\) into this object. Therefore we
have the answer for \(\mfk{C}\).

(Proposition 9)\(\varphi_S\) is a universally repelling object in \(\mfk{C}\).

## Principal and factorial ring

An ideal \(\mfk{o} \in A\) is said
to be **principal** if there exists some \(a \in A\) such that \(Aa = \mfk{o}\). For example for \(\mb{Z}\), the ideal \[
\{\cdots,-2,0,2,4,\cdots\}
\] is principal and we may write \(2\mb{Z}\). If every ideal of a
**commutative** ring \(A\)
is principal, we say \(A\) is
principal. Further we say \(A\) is a
**PID** if \(A\) is also
an integral domain (entire). When it comes to ring of fractions, we also
have the following proposition:

(Proposition 10)Let \(A\) be a principal ring and \(S\) a multiplicatively closed subset with \(0 \notin S\), then \(S^{-1}A\) is principal as well.

*Proof.* Let \(I \subset
S^{-1}A\) be an ideal. If \(a \in
S\) where \(a/s \in I\), then we
are done since then \((s/a)(a/s) = 1/1 \in
I\), which implies \(I=S^{-1}A\)
itself, hence we shall assume \(a \notin
S\) for all \(a/s \in I\). But
for \(a/s \in I\) we also have \((a/s)(s/1)=a/1 \in I\). Therefore \(J=\varphi_S^{-1}(I)\) is not empty. \(J\) is an ideal of \(A\) since for \(a
\in A\) and \(b \in J\), we have
\(\varphi_S(ab) =ab/1=(a/1)(b/1) \in
I\) which implies \(ab \in J\).
But since \(A\) is principal, there
exists some \(a\) such that \(Aa = J\). We shall discuss the relation
between \(S^{-1}A(a/1)\) and \(I\). For any \((c/u)(a/1)=ca/u \in S^{-1}A(a/1)\), clearly
we have \(ca/u \in I\), hence \(S^{-1}A(a/1)\subset I\). On the other hand,
for \(c/u \in I\), we see \(c/1=(c/u)(u/1) \in I\), hence \(c \in J\), and there exists some \(b \in A\) such that \(c = ba\), which gives \(c/u=ba/u=(b/u)(a/1) \in I\). Hence \(I \subset S^{-1}A(a/1)\), and we have
finally proved that \(I =
S^{-1}A(a/1)\). \(\square\)

As an immediate corollary, if \(A_\mfk{p}\) is the localization of \(A\) at \(\mfk{p}\), and if \(A\) is principal, then \(A_\mfk{p}\) is principal as well. Next we
go through another kind of rings. A ring is called
**factorial** (or a **unique factorization
ring** or **UFD**) if it is entire and if every
non-zero element has a unique factorization into irreducible elements.
An element \(a \neq 0\) is called
**irreducible** if it is not a unit and whenever \(a=bc\), then either \(b\) or \(c\) is a unit. For all non-zero elements in
a factorial ring, we have \[
a=u\prod_{i=1}^{r}p_i,
\] where \(u\) is a unit
(invertible).

In fact, every PID is a UFD (proof here).
Irreducible elements in a factorial ring is called **prime
elements** or simply **prime** (take \(\mathbb{Z}\) and prime numbers as an
example). Indeed, if \(A\) is a
factorial ring and \(p\) a prime
element, then \(Ap\) is a prime ideal.
But we are more interested in the ring of fractions of a factorial
ring.

(Proposition 11)Let \(A\) be a factorial ring and \(S\) a multiplicatively closed subset with \(0 \notin S\), then \(S^{-1}A\) is factorial.

*Proof.* Pick \(a/s \in
S^{-1}A\). Since \(A\) is
factorial, we have \(a=up_1 \cdots
p_k\) where \(p_i\) are primes
and \(u\) is a unit. But we have no
idea what are irreducible elements of \(S^{-1}A\). Naturally our first attack is
\(p_i/1\). And we have no need to
restrict ourselves to \(p_i\), we
should work on all primes of \(A\).
Suppose \(p\) is a prime of \(A\). If \(p \in
S\), then \(p/1 \in S\) is a
unit, not prime. If \(Ap \cap S \neq
\varnothing\), then \(rp \in S\)
for some \(r \in A\). But then \[
(p/1)(r/rp)=1,
\] again \(p/1\) is a unit, not
prime. Finally if \(Ap \cap S =
\varnothing\), then \(p/1\) is
prime in \(S^{-1}A\). For any \[
(a/s)(b/t)=ab/st=p/1,
\] we see \(ab=stp \not\in S\).
But this also gives \(ab \in Ap\) which
is a prime ideal, hence we can assume \(a \in
Ap\) and write \(a=rp\) for some
\(r \in A\). With this expansion we get
\[
ab=stp \implies rbp=stp \implies rb=st \implies (r/s)(b/t)=1/1.
\] Hence \(b/t\) is a unit,
\(p/1\) is a prime.

Conversely, suppose \(a/s\) is irreducible in \(S^{-1}A\). Since \(A\) is factorial, we may write \(a=u\prod_{i}p_i\). \(a\) cannot be an element of \(S\) since \(a/s\) is not a unit. We write \[ a/s=1/s[(u/1)(p_1/1)(p_2/1)\cdots(p_n/1)] \] We see there is some \(v \in A\) such that \(uv=1\) and accordingly \((u/1)(v/1)=uv/1=1/1\), hence \(u/1\) is a unit. We claim that there exist a unique \(p_k\) such that \(1 \leq k \leq n\) and \(Ap \cap S = \varnothing\). If not exists, then all \(p_j/1\) are units. If both \(p_{k}\) and \(p_{k'}\) satisfy the requirement and \(p_k \neq p_k'\), then we can write \(a/s\) as \[ a/s = \{1/s[(u/1)(p_1/1)\cdots(p_{k-1}/1)(p_{k+1}/1)\cdots(p_{k'-1}/1)(p_{k'+1}/1)\cdots(p_n/1)](p_k/1)\}(p_{k'}/1). \] Neither the one in curly bracket nor \(p_{k'}/1\) is unit, contradicting the fact that \(a/s\) is irreducible. Next we show that \(a/s=p_k/1\). For simplicity we write \[ b = u\prod_{i=1 \\ i \neq k}^{n} p_i, \quad a = bp_k. \] Note \(a/s = bp_k/s = (b/s)(p_k/1)\). Since \(a/s\) is irreducible, \(p_k/1\) is not a unit, we conclude that \(b/s\) is a unit. We are done for the study of irreducible elements of \(S^{-1}A\): it is of the form \(p/1\) (up to a unit) where \(p\) is prime in \(A\) and \(Ap \cap S = \varnothing\).

Now we are close to the fact that \(S^{-1}A\) is also factorial. For any \(a/s \in S^{-1}A\), we have an expansion \[ a/s=1/s[(u/1)(p_1/1)(p_2/1)\cdots(p_n/1)]. \] Let \(p'_1,p'_2,\cdots,p'_j\) be those whose generated prime ideal has nontrivial intersection with \(S\), then \(p'_1/1, p'_2/1,\cdots,p'_j/1\) are units of \(S^{-1}A\). Let \(q_1,q_2,\cdots,q_k\) be other \(p_i\)'s, then \(q_1/1,q_2/1,\cdots,q_k/1\) are irreducible in \(S^{-1}A\). This gives \[ a/s = [(1/s)(p'_1/1)(p'_2/1)\cdots(p'_j/1)]\prod_{i=1}^{k}(q_i/1). \] Hence \(S^{-1}A\) is factorial as well. \(\square\)

We finish the whole post by a comprehensive proposition:

(Proposition 12)Let \(A\) be a factorial ring and \(p\) a prime element, \(\mfk{p}=Ap\). The localization of \(A\) at \(\mfk{p}\) is principal.

*Proof.* For \(a/s \in
S^{-1}A\), we see \(p\) does not
divide \(s\) since if \(s = rp\) for some \(r \in A\), then \(s \in \mfk{p}\), contradicting the fact
that \(S = A \setminus \mfk{p}\). Since
\(A\) is factorial, we may write \(a = cp^n\) for some \(n \geq 0\) and \(p\) does not divide \(c\) as well (which gives \(c \in S\). Hence \(a/s = (c/s)(p^n/1)\). Note \((c/s)(s/c)=1/1\) and therefore \(c/s\) is a unit. For every \(a/s \in S^{-1}A\) we may write it as \[
a/s = u(p^n/1),
\] where \(u\) is a unit of
\(S^{-1}A\).

Let \(I\) be any ideal in \(S^{-1}A\), and \[ m = \min\{n:u(p^n/1) \in I, u \text{ is a unit }\}. \] Let's discuss the relation between \(S^{-1}A(p^m/1)\) and \(I\). First we see \(S^{-1}A(p^m/1)=S^{-1}A(up^m/1)\) since if \(v\) is the inverse of \(u\), we get \[ vS^{-1}A(up^m/1)=S^{-1}A(p^m/1) \subset S^{-1}A(up^m/1), \\ S^{-1}A(up^m/1)=uS^{-1}A(p^m/1)\subset S^{-1}A(p^m/1). \] Any element of \(S^{-1}A(up^m/1)\) is of the form \[ vup^{m+k}/1=v(p^k/1)up^m/1. \] Since \(up^m/1 \in I\), we see \(vup^{m+k}/1 \in I\) as well, hence \(S^{-1}A(up^m/1) \subset I\). On the other hand, any element of \(I\) is of the form \(wup^{m+n}/1=w(p^n/1)u(p^m/1)\) where \(w\) is a unit and \(n \geq 0\). This shows that \(vup^{m+n}/1 \in S^{-1}A(up^m/1)\). Hence \(S^{-1}A(p^m/1)=S^{-1}A(up^m/1)=I\) as we wanted. \(\square\)