field of fractions

field of fractions (plural fields of fractions)

1. (algebra, ring theory) The smallest field in which a given ring can be embedded.
   • 1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,

     The general construction of the field of fractions ${\displaystyle \mathbb {Q} _{R}}$ out of ${\displaystyle R}$ is an exact parallel of the construction of the field of rational numbers ${\displaystyle \mathbb {Q} }$ out of the ring of integers ${\displaystyle \mathbb {Z} }$.

   • 1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,

     In this no., A and B denote two integrally closed Noetherian domains such that A ⊂ B and B is a finitely generated A-module and K and L the fields of fractions of A and B respectively.

   • 2013, Jean-Paul Bézivin, Kamal Boussaf, Alain Escassut, “Some old and new results on the zeros of the derivative of a p-adic meromorphic function”, in Khodr Shamseddine, editor, Advances in Ultrametric Analysis: 12th International Conference on p-adic Functional Analysis, American Mathematical Society, page 23:

     We denote by ${\displaystyle {\mathcal {A}}(\mathbb {K} )}$ the ${\displaystyle \mathbb {K} }$-algebra of entire functions in ${\displaystyle \mathbb {K} }$ i.e. the set of power series with coefficients in ${\displaystyle \mathbb {K} }$ converging in all ${\displaystyle \mathbb {K} }$ and we denote by ${\displaystyle {\mathcal {M}}(\mathbb {K} )}$ the field of meromorphic functions in ${\displaystyle \mathbb {K} }$, i.e. the field of fractions of ${\displaystyle {\mathcal {A}}(\mathbb {K} )}$.

Loosely speaking, the minimal embedding field must include the inverse of each nonzero element of the original ring and all multiples of each inverse.

May be denoted Frac(R) or Quot(R).

The synonym quotient field risks confusion with quotient ring or quotient of a ring by an ideal, a quite different concept.

• field of quotients, fraction field, quotient field

• ring of fractions