nLab ring of fractions

Idea

In the same vein that commutative rings are to integral domains and GCD rings are to GCD domains, rings of fractions are to field of fractions.

Definition

Let $R$ be a commutative ring. A multiplicative submonoid of $R$ is a subset $S \subseteq R$ with an injection $m:S \hookrightarrow R$, an element $1_S \in S$ and a function $(-)\cdot_S(-):S \times S \to S$, such that $m(1_S) = 1$ and for all elements $a \in S$ and $b \in S$, $s(a \cdot_S b) = s(a) \cdot s(b)$. The localization of $R$ away from a multiplicative submonoid $S$ is the initial commutative ring $R[S^{-1}]$ with a commutative ring homomorphism $h:R \to R[S^{-1}]$ and a function $(-)^{-1}:S \to R[S^{-1}]$, such that for all $a \in S$, $h(m(a)) \cdot a^{-1} = 1$.

The ring of fractions $R[\mathrm{Reg}(R)^{-1}]$ is the localization of $R$ away from the multiplicative subset of regular elements $\mathrm{Reg}(R)$. Its elements are fractions $\frac{a}{b}$ where $a\in R$ and $b\in \mathrm{Reg}(R)$ which are by the definition the equivalence classes of pairs $(a,b) \in R\times \mathrm{Reg}(R)$ and $(a,b)\sim (c,d)$ iff $a \cdot d = b \cdot c$. The addition is given by the formula

$$\frac{a}{b}+\frac{c}{d} = \frac{a \cdot d+b \cdot c}{b \cdot d}$$

and multiplication by

$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

For $a \in R$, $b \in \mathrm{Reg}(R)$, $c \in \mathrm{Reg}(R)$, and $d \in \mathrm{Reg}(R)$, division is given by

$$\frac{a}{b} \div \frac{c}{d} = \frac{a \cdot d}{b \cdot c}$$

 Examples

• The ring of fractions of the integers is the rational numbers $\mathbb{Q} \coloneqq \mathbb{Z}[\mathrm{Reg}(\mathbb{Z})^{-1}]$.

• The ring of fractions of a discrete integral domain is a discrete field.

• The ring of fractions of a Heyting integral domain is a Heyting field.

• The ring of fractions of a strict approximate integral domain is a local ring.

• The ring of fractions of any commutative ring is a prefield ring.

Related concepts

• Grothendieck group

• localization

• prefield ring

• multiplicative submonoid of cancellative elements

References

• Michael Francis Atiyah, Ian Grant MacDonald. Introduction to Commutative Algebra. Addison-Wesley, 1969. (pdf, pdf)

• Frank Quinn, Proof Projects for Teachers (pdf)

• Max Zeuner, Univalent Foundations of Constructive Algebraic Geometry (arXiv:2407.17362)