Dictionary

field of fractions en · NOUN

Meanings

1. (qualifier: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 out of R is an exact parallel of the construction of the field of rational numbers ℚ out of the ring of integers ℤ.
   • We denote by #92;mathcal#123;A#125;(#92;mathbb#123;K#125;) the #92;mathbb#123;K#125;-algebra of entire functions in #92;mathbb#123;K#125; i.e. the set of power series with coefficients in #92;mathbb#123;K#125; converging in all #92;mathbb#123;K#125; and we denote by #92;mathcal#123;M#125;(#92;mathbb#123;K#125;) the field of meromorphic functions in #92;mathbb#123;K#125;, i.e. the field of fractions of #92;mathcal#123;A#125;(#92;mathbb#123;K#125;). 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:
   • 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.

Forms

Hypernyms

ring of fractions

Synonyms

field of quotients · fraction field · quotient field