Fields of fractions

Fields of fractions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 08 January 2012

Fields of fractions

Let A be a commutative ring. A zero divisor is an element aA such that there exists b0 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,bA and b0 } , 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 ζ:A𝕂 then there is a unique ring homomorphism φ:𝔽 𝕂 such that ζ=φ ι.
    A 𝔽 𝕂 ι ζ φ

Notes and References

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

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


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

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