nLab ring of fractions

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Context

Algebra

higher algebra

universal algebra

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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

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

References

Last revised on December 9, 2022 at 00:29:01. See the history of this page for a list of all contributions to it.