field of fractions

== English == === Noun === field of fractions (plural fields of fractions) (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 Q R {\displaystyle \mathbb {Q} _{R}} out of R {\displaystyle R} is an exact parallel of the construction of the field of rational numbers Q {\displaystyle \mathbb {Q} } out of the ring of integers Z {\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. ==== Usage notes ==== 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. ==== Synonyms ==== field of quotients, fraction field, quotient field ==== Hypernyms ==== ring of fractions