"integral element" meaning in English

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      "form": "integral elements",
      "tags": [
        "plural"
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  "lang_code": "en",
  "pos": "noun",
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        {
          "text": "1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,\nA subring of 𝔅 containing the ring o of integral elements of the field k_0(π), distinct from 𝔅, and not contained in any other subring of 𝔅 distinct from 𝔅, is called a maximal ring of the algebra 𝔅. In a division algebra, the only maximal ring is the ring of integral elements."
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,\nIf 𝔖 is the ring of integral elements in a commutative ring 𝔗 (over a subring R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖)."
        },
        {
          "ref": "2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:",
          "text": "In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^([Richard Dedekind's]) theory.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R."
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          "ring",
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          "extension ring",
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          "subring",
          "subring"
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          "root",
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          "monic",
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          "polynomial",
          "polynomial"
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          "coefficient",
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      "qualifier": "commutative algebra; ring theory; commutative algebra; ring theory",
      "raw_glosses": [
        "(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R."
      ],
      "related": [
        {
          "word": "algebraic integer"
        },
        {
          "word": "algebraic number"
        },
        {
          "word": "integral closure"
        },
        {
          "word": "integral extension"
        }
      ],
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        "algebra",
        "mathematics",
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      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "element of a given ring that is a root of some monic polynomial with coefficients in a given subring",
          "tags": [
            "masculine"
          ],
          "word": "elemento intero"
        }
      ],
      "wikipedia": [
        "integral element"
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    }
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  "word": "integral element"
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  "pos": "noun",
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      "word": "algebraic integer"
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    {
      "word": "algebraic number"
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    {
      "word": "integral closure"
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    {
      "word": "integral extension"
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          "text": "1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,\nA subring of 𝔅 containing the ring o of integral elements of the field k_0(π), distinct from 𝔅, and not contained in any other subring of 𝔅 distinct from 𝔅, is called a maximal ring of the algebra 𝔅. In a division algebra, the only maximal ring is the ring of integral elements."
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,\nIf 𝔖 is the ring of integral elements in a commutative ring 𝔗 (over a subring R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖)."
        },
        {
          "ref": "2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:",
          "text": "In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^([Richard Dedekind's]) theory.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R."
      ],
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        "(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R."
      ],
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "element of a given ring that is a root of some monic polynomial with coefficients in a given subring",
      "tags": [
        "masculine"
      ],
      "word": "elemento intero"
    }
  ],
  "word": "integral element"
}