"algebra over a field" meaning in English

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      "form": "algebras over a field",
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        "plural"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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          "name": "Algebra",
          "orig": "en:Algebra",
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      "glosses": [
        "A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)"
      ],
      "hypernyms": [
        {
          "word": "ring"
        },
        {
          "word": "vector space"
        }
      ],
      "hyponyms": [
        {
          "word": "associative algebra"
        },
        {
          "word": "division algebra"
        },
        {
          "word": "non-associative algebra"
        }
      ],
      "id": "en-algebra_over_a_field-en-noun-tWh1Dkyb",
      "links": [
        [
          "algebra",
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        [
          "vector space",
          "vector space"
        ],
        [
          "field",
          "field#English"
        ],
        [
          "bilinear",
          "bilinear#English"
        ],
        [
          "scalar multiplication",
          "scalar multiplication"
        ],
        [
          "distributivity",
          "distributivity"
        ],
        [
          "ring",
          "ring#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)"
      ],
      "related": [
        {
          "word": "algebra over a ring"
        }
      ],
      "synonyms": [
        {
          "word": "algebra"
        },
        {
          "word": "k-algebra"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "algebra over a field"
      ]
    }
  ],
  "word": "algebra over a field"
}

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    {
      "word": "ring"
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    {
      "word": "vector space"
    }
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  "hyponyms": [
    {
      "word": "associative algebra"
    },
    {
      "word": "division algebra"
    },
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      "word": "non-associative algebra"
    }
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "algebra over a ring"
    }
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      "glosses": [
        "A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)"
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        [
          "field",
          "field#English"
        ],
        [
          "bilinear",
          "bilinear#English"
        ],
        [
          "scalar multiplication",
          "scalar multiplication"
        ],
        [
          "distributivity",
          "distributivity"
        ],
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          "ring",
          "ring#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)"
      ],
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  "synonyms": [
    {
      "word": "algebra"
    },
    {
      "word": "k-algebra"
    }
  ],
  "word": "algebra over a field"
}