Fields of fractions

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

Last updates: 08 January 2012

Fields of fractions

Let $A$ be a commutative ring. A zero divisor is an element $a \in A$ such that there exists $b \neq 0$ with $a b = 0$ .

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

Let $A$ be an integral domain. The field of fractions of $A$ is the set $𝔽 = {\frac{a}{b} | a, b \in A and b \neq 0},$ with $\frac{a}{b} = \frac{c}{d} if a d = b c,$ and operations given by $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d} and \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d} .$

Let

A

be an integral domain. Let

𝔽

be the field of fractions over

A

The relation $=$ is an equivalence relation, the operations on $𝔽$ are well defined and $𝔽$ is a field.
The map $\begin{matrix} ι : & A & ⟶ & 𝔽 \\ a & ⟼ & \frac{a}{1} \end{matrix}$ is an injective homomorphism.
If $𝕂$ is a field with an injective ring homomorphism $ζ : A \to 𝕂$ then there is a unique ring homomorphism $φ : 𝔽 \to 𝕂$ such that $ζ = φ \circ ι$ .

Notes and References

Many curricula introduce the field of fractions in primary school, when calculations with fractions are introduced. The rational numbers $ℚ$ are the field of fraction of the integers $ℤ$ .

Part (c) of Theorem 1.1 is the universal property for fields of fractions.

References

[Bou] N. Bourbaki, Théorie des Ensembles, Chapter III, Masson, Springer-Verlag, 1970 MR??????

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