"fractional ideal" meaning in All languages combined

{
  "forms": [
    {
      "form": "fractional ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "fractional ideal (plural fractional ideals)",
      "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",
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          "source": "w"
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          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
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      "examples": [
        {
          "ref": "1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476",
          "text": "Products of fractional ideals are again fractional ideals, since if A#x5C;alpha#x5C;subseteqR and B#x5C;beta#x5C;subseteqR, then (AB)(#x5C;alpha#x5C;beta)#x5C;subseteqR.",
          "type": "quotation"
        },
        {
          "ref": "2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10",
          "text": "Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.",
          "type": "quotation"
        },
        {
          "text": "2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,\nA fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is\na⁻¹=x∈F;xa⊂𝒪_F.\n[…] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb."
        }
      ],
      "glosses": [
        "Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R."
      ],
      "id": "en-fractional_ideal-en-noun-Kvmm0X-t",
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          "algebra",
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          "integral domain",
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        ],
        [
          "submodule",
          "submodule"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R."
      ],
      "synonyms": [
        {
          "sense": "invertible ideal",
          "word": "R-submodule of Frac(R) such that for some nonzero r∈R"
        },
        {
          "sense": "invertible ideal",
          "word": "rI ⊆ R"
        }
      ],
      "topics": [
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        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "fractional ideal"
      ]
    }
  ],
  "word": "fractional ideal"
}

{
  "forms": [
    {
      "form": "fractional ideals",
      "tags": [
        "plural"
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  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
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        "en:Algebra"
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      "examples": [
        {
          "ref": "1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476",
          "text": "Products of fractional ideals are again fractional ideals, since if A#x5C;alpha#x5C;subseteqR and B#x5C;beta#x5C;subseteqR, then (AB)(#x5C;alpha#x5C;beta)#x5C;subseteqR.",
          "type": "quotation"
        },
        {
          "ref": "2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10",
          "text": "Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.",
          "type": "quotation"
        },
        {
          "text": "2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,\nA fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is\na⁻¹=x∈F;xa⊂𝒪_F.\n[…] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb."
        }
      ],
      "glosses": [
        "Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R."
      ],
      "links": [
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          "algebra",
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          "integral domain"
        ],
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          "field of fractions",
          "field of fractions"
        ],
        [
          "submodule",
          "submodule"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R."
      ],
      "topics": [
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      "wikipedia": [
        "fractional ideal"
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  "synonyms": [
    {
      "sense": "invertible ideal",
      "word": "R-submodule of Frac(R) such that for some nonzero r∈R"
    },
    {
      "sense": "invertible ideal",
      "word": "rI ⊆ R"
    }
  ],
  "word": "fractional ideal"
}