Fields of Fractions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 October 2009

Fields of fractions

Let $A$ be a commutative ring. A zero divisor is an element $a\in A$ such that $b\ne 0$ and $ab=0$.

An integral domain is a commutative ring $A$ with no zero divisors except $0$.

Let $A$ be an integral domain. A field of fractions of $A$ is the set $$𝔽=\left\{\frac{a}{b}|a,b\in A,b\ne 0\right\}\text{,}$$ with $$\frac{a}{b}=\frac{c}{d}\text{if}ad=bc\text{,}$$ and operations given by $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\text{and}\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}\text{.}$$

Let $A$ be an integral domain. Let $𝔽$ be the field of fractions over $A$.

1. The operations on $𝔽$ are well defined and $𝔽$ is a field.
2. The map $$\begin{array}{rcl}\iota :A& ⟶& 𝔽\\ a& ⟼& \frac{a}{1}\end{array}$$ is an injective ring homomorphism.
3. If $𝕂$ is a field with an injective ring homomorphism $\zeta :A\to 𝕂$ then there is a unique ring homomorphism $\varphi :𝔽\to 𝕂$ such that $\zeta =\varphi \circ \iota$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)