field of fractions

《近世代数》 · 第 140 页 · 定义 1

设 $R$ 是一个整环，如果有一个域 $F$ 使得从 $R$ 到 $F$ 有一个单的环同态 $\sigma$，并且 $F$ 的每一个元素都可以表示成 $\sigma(a)\sigma(b)^{-1}$ 的形式，其中 $a\in R$，$b\in R^*$，那么把 $F$ 称为 $R$ 的分式域。

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《Abstract Algebra》 · 第 261、263 页 · Section 7.5「Rings of Fractions」，Theorem 15 及 Definition

Theorem 15. Let $R$ be a commutative ring. Let $D$ be any nonempty subset of $R$ that does not contain $0$, does not contain any zero divisors and is closed under multiplication (i.e., $ab\in D$ for all $a,b\in D$). Then there is a commutative ring $Q$ with $1$ such that $Q$ contains $R$ as a subring and every element of $D$ is a unit in $Q$. The ring $Q$ has the following additional properties. (1) every element of $Q$ is of the form $rd^{-1}$ for some $r\in R$ and $d\in D$. In particular, if $D=R-\{0\}$ then $Q$ is a field. (2) (uniqueness of $Q$) The ring $Q$ is the “smallest” ring containing $R$ in which all elements of $D$ become units, in the following sense. Let $S$ be any commutative ring with identity and let $\varphi:R\to S$ be any injective ring homomorphism such that $\varphi(d)$ is a unit in $S$ for every $d\in D$. Then there is an injective homomorphism $\Phi:Q\to S$ such that $\Phi|_R=\varphi$. In other words, any ring containing an isomorphic copy of $R$ in which all the elements of $D$ become units must also contain an isomorphic copy of $Q$. […] Definition. Let $R$, $D$ and $Q$ be as in Theorem 15. (1) The ring $Q$ is called the ring of fractions of $D$ with respect to $R$ and is denoted $D^{-1}R$. (2) If $R$ is an integral domain and $D=R-\{0\}$, $Q$ is called the field of fractions or quotient field of $R$.