"field of fractions" meaning in All languages combined

{
  "forms": [
    {
      "form": "fields of fractions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "field of fractions (plural fields of fractions)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,\nThe 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."
        },
        {
          "text": "1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,\nIn 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."
        },
        {
          "ref": "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",
          "text": "We denote by #x5C;mathcal#x7B;A#x7D;(#x5C;mathbb#x7B;K#x7D;) the #x5C;mathbb#x7B;K#x7D;-algebra of entire functions in #x5C;mathbb#x7B;K#x7D; i.e. the set of power series with coefficients in #x5C;mathbb#x7B;K#x7D; converging in all #x5C;mathbb#x7B;K#x7D; and we denote by #x5C;mathcal#x7B;M#x7D;(#x5C;mathbb#x7B;K#x7D;) the field of meromorphic functions in #x5C;mathbb#x7B;K#x7D;, i.e. the field of fractions of #x5C;mathcal#x7B;A#x7D;(#x5C;mathbb#x7B;K#x7D;).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The smallest field in which a given ring can be embedded."
      ],
      "hypernyms": [
        {
          "word": "ring of fractions"
        }
      ],
      "id": "en-field_of_fractions-en-noun-vrO681gC",
      "links": [
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          "algebra",
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          "field",
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        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "embed",
          "embed"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) The smallest field in which a given ring can be embedded."
      ],
      "synonyms": [
        {
          "word": "field of quotients"
        },
        {
          "word": "fraction field"
        },
        {
          "word": "quotient field"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "field of fractions"
      ]
    }
  ],
  "word": "field of fractions"
}

{
  "forms": [
    {
      "form": "fields of fractions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "fields of fractions"
      },
      "expansion": "field of fractions (plural fields of fractions)",
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  ],
  "hypernyms": [
    {
      "word": "ring of fractions"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
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        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
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      ],
      "examples": [
        {
          "text": "1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,\nThe 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."
        },
        {
          "text": "1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,\nIn 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."
        },
        {
          "ref": "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",
          "text": "We denote by #x5C;mathcal#x7B;A#x7D;(#x5C;mathbb#x7B;K#x7D;) the #x5C;mathbb#x7B;K#x7D;-algebra of entire functions in #x5C;mathbb#x7B;K#x7D; i.e. the set of power series with coefficients in #x5C;mathbb#x7B;K#x7D; converging in all #x5C;mathbb#x7B;K#x7D; and we denote by #x5C;mathcal#x7B;M#x7D;(#x5C;mathbb#x7B;K#x7D;) the field of meromorphic functions in #x5C;mathbb#x7B;K#x7D;, i.e. the field of fractions of #x5C;mathcal#x7B;A#x7D;(#x5C;mathbb#x7B;K#x7D;).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The smallest field in which a given ring can be embedded."
      ],
      "links": [
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        "(algebra, ring theory) The smallest field in which a given ring can be embedded."
      ],
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        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "field of fractions"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "field of quotients"
    },
    {
      "word": "fraction field"
    },
    {
      "word": "quotient field"
    }
  ],
  "word": "field of fractions"
}