"free abelian group" meaning in English

{
  "forms": [
    {
      "form": "free abelian groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "head": "free abelian group"
      },
      "expansion": "free abelian group (plural free abelian groups)",
      "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"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "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"
        }
      ],
      "id": "en-free_abelian_group-en-noun-ll5Y9Ncz",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "free module",
          "free module"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "integer",
          "integer"
        ]
      ],
      "raw_glosses": [
        "(algebra) a free module over the ring of integers"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "free abelian group"
      ]
    }
  ],
  "word": "free abelian group"
}

{
  "forms": [
    {
      "form": "free abelian groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "head": "free abelian group"
      },
      "expansion": "free abelian group (plural free abelian groups)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "free module"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "en:Algebra"
      ],
      "examples": [
        {
          "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"
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "free module",
          "free module"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "integer",
          "integer"
        ]
      ],
      "raw_glosses": [
        "(algebra) a free module over the ring of integers"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "free abelian group"
      ]
    }
  ],
  "word": "free abelian group"
}