# 代數導論二 Week 5 (Part 3) - Field of Fraction

[TOC]

## 定義：

:::warning

Suppose that $R$ is an integral domain, let

$$
D = R \setminus \{0\}
$$

Let

$$
S = \{(a, b) \mid a \in R, b \in D\}
$$

:::

## 定義：Fraction

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Define a binary relation $\sim$ on $S$ by

$$
(a_1, b_1) \sim (a_2, b_2) \iff a_1b_2 = b_1a_2
$$

Observe that $\sim$ is an equivalent relation. Define

$$
\text{Frac}(R) = S /\sim
$$

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## 定義：加法

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Define addition on $[(a, b)], [(c, d)] \in \text{Frac}(R)$ as:

$$
[(a, b)] + [(c, d)] = [(ad + bc, bd)]
$$

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### 觀察

:::danger

1. Both addition and production are well-defined
2. Addition is commutative
3. Addition os associative
4. $[(0, 1)]$ is the identity for addition.
5. $[(-a, b)] + [(a, b)] = [(0, 1)]$

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## 定義：乘法

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Define production on $[(a, b)], [(c, d)] \in \text{Frac}(R)$ as:

$$
[(a, b)] \cdot [(c, d)] = [(ac, bd)]
$$

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### 觀察：

:::danger

1. Multiplication is well-defined
2. Multipliation is commutative
3. Multiplication is associative
4. Distribution law
5. $[(1, 1)]$ is the identity of multiplication

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### 觀察：所有元素都有乘法反元素

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Suppose that

$$
[(a, b)] \in \text{Frac}(R) \setminus\{0\}
$$

Then

$$
[(a, b)] \cdot [b, a) = 1_{\text{Frac}(R)}
$$

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## 定義：Field of Fraction

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$\text{Frac}(R)$ is called the field of fractions of $R$

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## 定理：Universal Property

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Suppose that $Q = \text{Frac}(R)$. Let $F$ be a field and

$$
i : R \to \mathcal F
$$

an injective ring homomorphism, then $i$ factor through $Q$ i.e. let

$$
\begin{align}
c : R &\to Q
\newline
r &\mapsto [(r, 1)]
\end{align}
$$

Then there exists unique unital homomorphism $\tilde i$:

$$
\begin{align}
\tilde i: Q &\to \mathcal F
\end{align}
$$

Such that

$$
i = c \circ \tilde i
$$

Or in commutative diagram:

$$
\require{AMScd}
\begin{CD}
R @>c>> {\text{Frac}{(F)}}\\
@. {\searrow} @VV {\tilde i} V\\
@.F
\end{CD}
$$

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Let

$$
[(a, b)] \to i(a)i(b)^{-1}
$$

Suppose that $[(a, b)] \sim [(a', b')]$, then

$$
i(ab)
$$

### 推論：

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Let $R$ be a integral domain. Then every fields containing $R$ contains $\text{Frac}(R)$

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