"free abelian group" meaning in All languages combined

{
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    {
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  "lang_code": "en",
  "pos": "noun",
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        {
          "text": "A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ=⨁ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)"
        }
      ],
      "glosses": [
        "A free module over the ring of integers."
      ],
      "hypernyms": [
        {
          "word": "free module"
        }
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      "id": "en-free_abelian_group-en-noun-coy~K5iE",
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          "algebra",
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          "free module",
          "free module"
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          "ring",
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          "integer",
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      "raw_glosses": [
        "(algebra) A free module over the ring of integers."
      ],
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  "word": "free abelian group"
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        {
          "text": "A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ=⨁ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)"
        }
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        "(algebra) A free module over the ring of integers."
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