Ring of fractions

Section 6.2 Ring of fractions

Lemma 6.2.1.

Let \(A\) be a commutative ring and let \(S\) be a multiplicatively closed subset of \(A\text{.}\) Consider

\begin{equation*} A\times S=\{(a,s):a\in A\text{ and }s\in S\}\text{.} \end{equation*}

Define a relation ‘\(\sim\)’ on \(\mathcal{S}\) as follows:

\begin{equation*} (a,s)\sim (b,t)\quad\text{if there exists}\,u\in S\,\text{such that}\,u(at-bs)=0\in A. \end{equation*}

This is an equivalence relation.

Convention 6.2.2.

We keep the notation of Lemma 6.2.1. We denote the equivalence class of \((a,s)\) by \(a/s\text{.}\)

Proposition 6.2.3.

We keep the notations of Lemma 6.2.1. The set

\begin{equation*} S^{-1}A=\left\{a/s:a\in A\,\text{and}\,s\in S\right\} \end{equation*}

with addition and multiplication defined below is a commutative ring.

\((a/s)+(b/t)=(at+bs)/st\) for \(a,b\in A\) and \(s,t\in S\)
\((a/s)\cdot (b/s)=ab/st\) for \(a,b\in A\) and \(s,t\in S\text{.}\)
The additive identity of \(S^{-1}A\) is \(0_A/1_A\) while the multiplicative identity of \(S^{-1}A\) is \(1_A/1_A\text{.}\)

Definition 6.2.4. (Ring of fractions).

Let \(A\) be a commutative ring and let \(S\) be a multiplicatively closed subset of \(A\text{.}\) Then

\begin{equation*} S^{-1}A=\left\{a/s:a\in A\,\text{and}\,s\in S\right\} \end{equation*}

is called the ring of fractions of \(A\) with respect to \(S\text{.}\)

Example 6.2.5. (Localization at a prime).

Let \(A\) be a commutative ring and let \(\mfp\) be a prime ideal in \(A\text{.}\) Consider a multiplicative set \(S=A\setminus\mfp\text{.}\) The ring of fractions \(S^{-1}A\) is also called the localization at a prime \(\mfp\). It is often denoted by \(A_{\mfp}\text{.}\)

Example 6.2.6. (Field of fractions/Quotient field).

Let \(A\) be an integral domain. The set \(S=A\setminus\{0\}\) is a multiplicatively closed subset of \(A\text{.}\) The ring \(S^{-1}A\) is a field and it is called the field of fractions of \(A\) or the quotient field of \(A\). Note that it is a particular case of Example 6.2.5 with the prime ideal \((0)\text{.}\)

The field of rational numbers \(\Q\) is obtained as the field of fractions of \(\Z\text{.}\)

Lemma 6.2.7.

Let \(A\) be an integral domain and let \(Q\) be the field of fractions of \(A\text{.}\) The mapping \(i_A\colon A\to Q\) given by \(a\mapsto a/1_A\) is a ring monomorphism.

Proposition 6.2.8. (Universal property of field of fractions).

Let \(A\) be an integral domain and let \(Q\) be the field of fractions of \(A\text{.}\) Suppose that \(K\) is a field. If there is a ring monomorphism \(f\colon A\to K\) then there is a unique ring monomorphism \(g\colon Q\to K\) such that \(f=g\circ i_A\text{.}\)

Proof.

We define \(g\colon Q\to K\) by \(a/b\mapsto f(a)f(b)^{-1}\text{.}\)

[Well-definedness].
Suppose that \(a/b=c/d\text{,}\) i.e., there exists \(e\in A\setminus\{0\}\) such that \(e(ad-bc)=0\in A\) (refer Lemma 6.2.1). Since \(A\) is an integral domain, \(ad=bc\text{.}\) Thus, \(f(a)f(d)=f(b)f(c)\text{.}\) By the assumption \(f\) is an monomorphism, \(b\neq 0\) and \(c\neq 0\text{.}\) Hence, \(f(a)f(b)^{-1}=f(c)f(d)^{-1}\text{.}\)
[Ring homomorphism].
Left to the reader.
[Injectivity].
Suppose that \(g(a/b)=f(a)f(b)^{-1}=0\text{.}\) Thus \(f(a)=0\) and since \(f\) is given to be a monomorphism \(a=0_A\text{.}\) By Lemma 6.2.1 we have \(0_A/b=0_A/1_A\) and thus \(g\) is injective.
[Uniqueness].
Let \(h\colon Q\to K\) be a ring monomorphism such that \(f=h\circ i_A\text{.}\) Thus \(f(a)=h(a/1_A)=g(a/1_A)\text{.}\) For \(0\neq b\in A\) we have
\begin{equation*} 1_K=h(1_A/1_A)=h(b/1_A\cdot 1_A/b)=h(b/1_A)\cdot h(1_A/b)=f(b)\cdot h(1_A/b). \end{equation*}
Hence \(h(1_A/b)=f(b)^{-1}\text{.}\) Therefore, \(h(a/b)=f(a)f(b)^{-1}\text{.}\)

Corollary 6.2.9.

If \(K\) is a field of characteristic zero then there is a unique ring monomorphism \(\Q\to K\text{.}\)
If \(K\) is a field of characteristic \(p\) then there is a unique ring monomorphism \(\mathbb{F}_p\to K\text{.}\)

Checkpoint 6.2.10. (Rational function field).

Consider the polynomial ring over a field \(F[x]\text{.}\) Describe the field of fractions/quotient field of \(F[x]\text{.}\) We denote the field of fractions of \(F[x]\) by \(F(x)\text{,}\) and it is called the rational function field.

Algebra-II: Companion notes

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