Fields of fractions

## Fields of fractions

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 $b\ne 0$ with $ab=0$.

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 $𝔽={ ab | a,b∈A and b≠0 } ,$ with $ab = cd if ad=bc,$ and operations given by $ab + cd = ad+bc bd and ab ⋅ cd = ac bd .$

Let $A$ be an integral domain. Let $𝔽$ be the field of fractions over $A$.
1. The relation $=$ is an equivalence relation, the operations on $𝔽$ are well defined and $𝔽$ is a field.
2. The map $ι: A ⟶ 𝔽 a ⟼ a1$ is an injective homomorphism.
3. If $𝕂$ is a field with an injective ring homomorphism $\zeta :A\to 𝕂$ then there is a unique ring homomorphism $\phi :𝔽\to 𝕂$ such that $\zeta =\phi \circ \iota$.

## 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??????